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(jax-301-custom-jvp-vjp)=
# Custom JVPs or VJPs with `custom_jvp` and `custom_vjp`

<!--* freshness: { reviewed: '2025-12-10' } *-->

There are two ways to define differentiation rules in JAX:

1. using [`jax.custom_jvp`](https://docs.jax.dev/en/latest/_autosummary/jax.custom_jvp.html) and [`jax.custom_vjp`](https://docs.jax.dev/en/latest/_autosummary/jax.custom_vjp.html) to define custom differentiation rules for Python functions that are already JAX-transformable; and
2. defining new `core.Primitive` instances along with all their transformation rules, for example to call into functions from other systems like solvers, simulators, or general numerical computing systems.

This notebook is about #1. To read instead about #2, see {doc}`custom-derivatives` and {doc}`hijax-types`.

Hijax primitives (see {doc}`custom-derivatives`) unify these two approaches, and are the recommended path going forward: a hijax primitive carries a JAX-traceable Python implementation along with custom rules for differentiation and other transformations. The APIs here remain fully supported, and are often the most convenient for simple cases.

For an introduction to JAX's automatic differentiation API, see {doc}`cookbook`. This notebook assumes some familiarity with [jax.jvp](https://docs.jax.dev/en/latest/_autosummary/jax.jvp.html) and [jax.grad](https://docs.jax.dev/en/latest/_autosummary/jax.grad.html), and the mathematical meaning of JVPs and VJPs.

+++

### TL;DR: Custom JVPs with `jax.custom_jvp`

```{code-cell}
import jax.numpy as jnp
from jax import custom_jvp

@custom_jvp
def f(x, y):
  return jnp.sin(x) * y

@f.defjvp
def f_jvp(primals, tangents):
  x, y = primals
  x_dot, y_dot = tangents
  primal_out = f(x, y)
  tangent_out = jnp.cos(x) * x_dot * y + jnp.sin(x) * y_dot
  return primal_out, tangent_out
```

```{code-cell}
from jax import jvp, grad

print(f(2., 3.))
y, y_dot = jvp(f, (2., 3.), (1., 0.))
print(y)
print(y_dot)
print(grad(f)(2., 3.))
```

```{code-cell}
# Equivalent alternative using the defjvps convenience wrapper

@custom_jvp
def f(x, y):
  return jnp.sin(x) * y

f.defjvps(lambda x_dot, primal_out, x, y: jnp.cos(x) * x_dot * y,
          lambda y_dot, primal_out, x, y: jnp.sin(x) * y_dot)
```

```{code-cell}
print(f(2., 3.))
y, y_dot = jvp(f, (2., 3.), (1., 0.))
print(y)
print(y_dot)
print(grad(f)(2., 3.))
```

### TL;DR: Custom VJPs with `jax.custom_vjp`

```{code-cell}
from jax import custom_vjp

@custom_vjp
def f(x, y):
  return jnp.sin(x) * y

def f_fwd(x, y):
  # Returns primal output and residuals to be used in backward pass by f_bwd.
  return f(x, y), (jnp.cos(x), jnp.sin(x), y)

def f_bwd(res, g):
  cos_x, sin_x, y = res # Gets residuals computed in f_fwd
  return (cos_x * g * y, sin_x * g)

f.defvjp(f_fwd, f_bwd)
```

```{code-cell}
print(grad(f)(2., 3.))
```

## Example problems

To get an idea of what problems `jax.custom_jvp` and `jax.custom_vjp` are meant to solve, let's go over one example for each — a `custom_jvp` for numerical stability, and a `custom_vjp` for gradient clipping. (More example problems, including enforcing a differentiation convention at a boundary and efficient implicit differentiation of fixed points, are worked in {doc}`custom-derivatives`; each translates directly to these APIs.) A more thorough introduction to the `jax.custom_jvp` and `jax.custom_vjp` APIs is in the next section.

+++

### Example: Numerical stability

One application of `jax.custom_jvp` is to improve the numerical stability of differentiation.

+++

Say we want to write a function called `log1pexp`, which computes $x \mapsto \log ( 1 + e^x )$. We can write that using `jax.numpy`:

```{code-cell}
def log1pexp(x):
  return jnp.log(1. + jnp.exp(x))

log1pexp(3.)
```

Since it's written in terms of `jax.numpy`, it's JAX-transformable:

```{code-cell}
from jax import jit, grad, vmap

print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
```

But there's a numerical stability problem lurking here:

```{code-cell}
print(grad(log1pexp)(100.))
```

That doesn't seem right! After all, the derivative of $x \mapsto \log (1 + e^x)$ is $x \mapsto \frac{e^x}{1 + e^x}$, and so for large values of $x$ we'd expect the value to be about 1.

We can get a bit more insight into what's going on by looking at the jaxpr for the gradient computation:

```{code-cell}
jit(grad(log1pexp)).trace(100.).jaxpr
```

Stepping through how the jaxpr would be evaluated, we can see that the last line would involve multiplying values that floating point math will round to 0 and $\infty$, respectively, which is never a good idea. That is, we're effectively evaluating `lambda x: (1 / (1 + jnp.exp(x))) * jnp.exp(x)` for large `x`, which effectively turns into `0. * jnp.inf`.

Instead of generating such large and small values, hoping for a cancellation that floats can't always provide, we'd rather just express the derivative function as a more numerically stable program. In particular, we can write a program that more closely evaluates the equal mathematical expression $1 - \frac{1}{1 + e^x}$, with no cancellation in sight.

This problem is interesting because even though our definition of `log1pexp` could already be JAX-differentiated (and transformed with [`jit`](https://docs.jax.dev/en/latest/_autosummary/jax.jit.html), [`vmap`](https://docs.jax.dev/en/latest/_autosummary/jax.vmap.html), ...), we're not happy with the result of applying standard autodiff rules to the primitives comprising `log1pexp` and composing the result. Instead, we'd like to specify how the whole function `log1pexp` should be differentiated, as a unit, and thus arrange those exponentials better.

This is one application of custom derivative rules for Python functions that are already JAX transformable: specifying how a composite function should be differentiated, while still using its original Python definition for other transformations (like `jit`, `vmap`, ...).

Here's a solution using `jax.custom_jvp`:

```{code-cell}
from jax import custom_jvp

@custom_jvp
def log1pexp(x):
  return jnp.log(1. + jnp.exp(x))

@log1pexp.defjvp
def log1pexp_jvp(primals, tangents):
  x, = primals
  x_dot, = tangents
  ans = log1pexp(x)
  ans_dot = (1 - 1/(1 + jnp.exp(x))) * x_dot
  return ans, ans_dot
```

```{code-cell}
print(grad(log1pexp)(100.))
```

```{code-cell}
print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
```

Here's a [`defjvps`](https://docs.jax.dev/en/latest/_autosummary/jax.custom_jvp.defjvps.html) convenience wrapper to express the same thing:

```{code-cell}
@custom_jvp
def log1pexp(x):
  return jnp.log(1. + jnp.exp(x))

log1pexp.defjvps(lambda t, ans, x: (1 - 1/(1 + jnp.exp(x))) * t)
```

```{code-cell}
print(grad(log1pexp)(100.))
print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
```

### Example: Gradient clipping

While in some cases we want to express a mathematical differentiation computation, in other cases we may even want to take a step away from mathematics to adjust the computation autodiff performs. One canonical example is reverse-mode gradient clipping.

For gradient clipping, we can use [`jnp.clip`](https://docs.jax.dev/en/latest/_autosummary/jax.numpy.clip.html) together with a [`jax.custom_vjp`](https://docs.jax.dev/en/latest/_autosummary/jax.custom_vjp.html) reverse-mode-only rule:

```{code-cell}
from functools import partial
from jax import custom_vjp

@custom_vjp
def clip_gradient(lo, hi, x):
  return x  # identity function

def clip_gradient_fwd(lo, hi, x):
  return x, (lo, hi)  # save bounds as residuals

def clip_gradient_bwd(res, g):
  lo, hi = res
  return (None, None, jnp.clip(g, lo, hi))  # use None to indicate zero cotangents for lo and hi

clip_gradient.defvjp(clip_gradient_fwd, clip_gradient_bwd)
```

```{code-cell}
import matplotlib.pyplot as plt
from jax import vmap

t = jnp.linspace(0, 10, 1000)

plt.plot(jnp.sin(t))
plt.plot(vmap(grad(jnp.sin))(t))
```

```{code-cell}
def clip_sin(x):
  x = clip_gradient(-0.75, 0.75, x)
  return jnp.sin(x)

plt.plot(clip_sin(t))
plt.plot(vmap(grad(clip_sin))(t))
```

## Basic usage of `jax.custom_jvp` and `jax.custom_vjp` APIs

+++

### Use `jax.custom_jvp` to define forward-mode (and, indirectly, reverse-mode) rules

Here's a canonical basic example of using [`jax.custom_jvp`](https://docs.jax.dev/en/latest/_autosummary/jax.custom_jvp.html), where the comments use
[Haskell-like type signatures](https://wiki.haskell.org/Type_signature):

```{code-cell}
from jax import custom_jvp
import jax.numpy as jnp

# f :: a -> b
@custom_jvp
def f(x):
  return jnp.sin(x)

# f_jvp :: (a, T a) -> (b, T b)
def f_jvp(primals, tangents):
  x, = primals
  t, = tangents
  return f(x), jnp.cos(x) * t

f.defjvp(f_jvp)
```

```{code-cell}
from jax import jvp

print(f(3.))

y, y_dot = jvp(f, (3.,), (1.,))
print(y)
print(y_dot)
```

In words, we start with a primal function `f` that takes inputs of type `a` and produces outputs of type `b`. We associate with it a JVP rule function `f_jvp` that takes a pair of inputs representing the primal inputs of type `a` and the corresponding tangent inputs of type `T a`, and produces a pair of outputs representing the primal outputs of type `b` and tangent outputs of type `T b`. The tangent outputs should be a linear function of the tangent inputs.

+++

You can also use `f.defjvp` as a decorator, as in

```python
@custom_jvp
def f(x):
  ...

@f.defjvp
def f_jvp(primals, tangents):
  ...
```

+++

Even though we defined only a JVP rule and no VJP rule, we can use both forward- and reverse-mode differentiation on `f`. JAX will automatically transpose the linear computation on tangent values from our custom JVP rule, computing the VJP as efficiently as if we had written the rule by hand:

```{code-cell}
from jax import grad

print(grad(f)(3.))
print(grad(grad(f))(3.))
```

For automatic transposition to work, the JVP rule's output tangents must be linear as a function of the input tangents. Otherwise a transposition error is raised.

+++

Multiple arguments work like this:

```{code-cell}
@custom_jvp
def f(x, y):
  return x ** 2 * y

@f.defjvp
def f_jvp(primals, tangents):
  x, y = primals
  x_dot, y_dot = tangents
  primal_out = f(x, y)
  tangent_out = 2 * x * y * x_dot + x ** 2 * y_dot
  return primal_out, tangent_out
```

```{code-cell}
print(grad(f)(2., 3.))
```

The [`defjvps`](https://docs.jax.dev/en/latest/_autosummary/jax.custom_jvp.defjvps.html) convenience wrapper lets us define a JVP for each argument separately, and the results are computed separately then summed:

```{code-cell}
@custom_jvp
def f(x):
  return jnp.sin(x)

f.defjvps(lambda t, ans, x: jnp.cos(x) * t)
```

```{code-cell}
print(grad(f)(3.))
```

Here's a `defjvps` example with multiple arguments:

```{code-cell}
@custom_jvp
def f(x, y):
  return x ** 2 * y

f.defjvps(lambda x_dot, primal_out, x, y: 2 * x * y * x_dot,
          lambda y_dot, primal_out, x, y: x ** 2 * y_dot)
```

```{code-cell}
print(grad(f)(2., 3.))
print(grad(f, 0)(2., 3.))  # same as above
print(grad(f, 1)(2., 3.))
```

As a shorthand, with `defjvps` you can pass a `None` value to indicate that the JVP for a particular argument is zero:

```{code-cell}
@custom_jvp
def f(x, y):
  return x ** 2 * y

f.defjvps(lambda x_dot, primal_out, x, y: 2 * x * y * x_dot,
          None)
```

```{code-cell}
print(grad(f)(2., 3.))
print(grad(f, 0)(2., 3.))  # same as above
print(grad(f, 1)(2., 3.))
```

Calling a `jax.custom_jvp` function with keyword arguments, or writing a `jax.custom_jvp` function definition with default arguments, are both allowed so long as they can be unambiguously mapped to positional arguments based on the function signature retrieved by the standard library `inspect.signature` mechanism.

+++

When you're not performing differentiation, the function `f` is called just as if it weren't decorated by `jax.custom_jvp`:

```{code-cell}
@custom_jvp
def f(x):
  print('called f!')  # a harmless side-effect
  return jnp.sin(x)

@f.defjvp
def f_jvp(primals, tangents):
  print('called f_jvp!')  # a harmless side-effect
  x, = primals
  t, = tangents
  return f(x), jnp.cos(x) * t
```

```{code-cell}
from jax import vmap, jit

print(f(3.))
```

```{code-cell}
print(vmap(f)(jnp.arange(3.)))
print(jit(f)(3.))
```

The custom JVP rule is invoked during differentiation, whether forward or reverse:

```{code-cell}
y, y_dot = jvp(f, (3.,), (1.,))
print(y_dot)
```

```{code-cell}
print(grad(f)(3.))
```

Notice that `f_jvp` calls `f` to compute the primal outputs. In the context of higher-order differentiation, each application of a differentiation transform will use the custom JVP rule if and only if the rule calls the original `f` to compute the primal outputs. (This represents a kind of fundamental tradeoff, where we can't make use of intermediate values from the evaluation of `f` in our rule _and also_ have the rule apply in all orders of higher-order differentiation.)

```{code-cell}
grad(grad(f))(3.)
```

You can use Python control flow with `jax.custom_jvp`:

```{code-cell}
@custom_jvp
def f(x):
  if x > 0:
    return jnp.sin(x)
  else:
    return jnp.cos(x)

@f.defjvp
def f_jvp(primals, tangents):
  x, = primals
  x_dot, = tangents
  ans = f(x)
  if x > 0:
    return ans, 2 * x_dot
  else:
    return ans, 3 * x_dot
```

```{code-cell}
print(grad(f)(1.))
print(grad(f)(-1.))
```

### Use `jax.custom_vjp` to define custom reverse-mode-only rules

While `jax.custom_jvp` suffices for controlling both forward- and, via JAX's automatic transposition, reverse-mode differentiation behavior, in some cases we may want to directly control a VJP rule, for example in the latter two example problems presented above. We can do that with [`jax.custom_vjp`](https://docs.jax.dev/en/latest/_autosummary/jax.custom_vjp.html):

```{code-cell}
from jax import custom_vjp
import jax.numpy as jnp

# f :: a -> b
@custom_vjp
def f(x):
  return jnp.sin(x)

# f_fwd :: a -> (b, c)
def f_fwd(x):
  return f(x), jnp.cos(x)

# f_bwd :: (c, CT b) -> CT a
def f_bwd(cos_x, y_bar):
  return (cos_x * y_bar,)

f.defvjp(f_fwd, f_bwd)
```

```{code-cell}
from jax import grad

print(f(3.))
print(grad(f)(3.))
```

In words, we again start with a primal function `f` that takes inputs of type `a` and produces outputs of type `b`. We associate with it two functions, `f_fwd` and `f_bwd`, which describe how to perform the forward- and backward-passes of reverse-mode autodiff, respectively.

The function `f_fwd` describes the forward pass, not only the primal computation but also what values to save for use on the backward pass. Its input signature is just like that of the primal function `f`, in that it takes a primal input of type `a`. But as output it produces a pair, where the first element is the primal output `b` and the second element is any "residual" data of type `c` to be stored for use by the backward pass. (This second output is analogous to [PyTorch's save_for_backward mechanism](https://pytorch.org/tutorials/beginner/examples_autograd/two_layer_net_custom_function.html).)

The function `f_bwd` describes the backward pass. It takes two inputs, where the first is the residual data of type `c` produced by `f_fwd` and the second is the output cotangents of type `CT b` corresponding to the output of the primal function. It produces an output of type `CT a` representing the cotangents corresponding to the input of the primal function. In particular, the output of `f_bwd` must be a sequence (e.g. a tuple) of length equal to the number of arguments to the primal function.

+++

So multiple arguments work like this:

```{code-cell}
from jax import custom_vjp

@custom_vjp
def f(x, y):
  return jnp.sin(x) * y

def f_fwd(x, y):
  return f(x, y), (jnp.cos(x), jnp.sin(x), y)

def f_bwd(res, g):
  cos_x, sin_x, y = res
  return (cos_x * g * y, sin_x * g)

f.defvjp(f_fwd, f_bwd)
```

```{code-cell}
print(grad(f)(2., 3.))
```

Calling a `jax.custom_vjp` function with keyword arguments, or writing a `jax.custom_vjp` function definition with default arguments, are both allowed so long as they can be unambiguously mapped to positional arguments based on the function signature retrieved by the standard library `inspect.signature` mechanism.

+++

As with `jax.custom_jvp`, the custom VJP rule comprised by `f_fwd` and `f_bwd` is not invoked if differentiation is not applied. If function is evaluated, or transformed with `jit`, `vmap`, or other non-differentiation transformations, then only `f` is called.

```{code-cell}
@custom_vjp
def f(x):
  print("called f!")
  return jnp.sin(x)

def f_fwd(x):
  print("called f_fwd!")
  return f(x), jnp.cos(x)

def f_bwd(cos_x, y_bar):
  print("called f_bwd!")
  return (cos_x * y_bar,)

f.defvjp(f_fwd, f_bwd)
```

```{code-cell}
print(f(3.))
```

```{code-cell}
print(grad(f)(3.))
```

```{code-cell}
from jax import vjp

y, f_vjp = vjp(f, 3.)
print(y)
```

```{code-cell}
print(f_vjp(1.))
```

**Forward-mode autodiff cannot be used on the** `jax.custom_vjp` **function** and will raise an error:

```{code-cell}
from jax import jvp

try:
  jvp(f, (3.,), (1.,))
except TypeError as e:
  print('ERROR! {}'.format(e))
```

If you want to use both forward- and reverse-mode, use `jax.custom_jvp` instead.

+++

We can use `jax.custom_vjp` together with `pdb` to insert a debugger trace in the backward pass:

```{code-cell}
import pdb

@custom_vjp
def debug(x):
  return x  # acts like identity

def debug_fwd(x):
  return x, x

def debug_bwd(x, g):
  pdb.set_trace()
  return g

debug.defvjp(debug_fwd, debug_bwd)
```

```{code-cell}
def foo(x):
  y = x ** 2
  y = debug(y)  # insert pdb in corresponding backward pass step
  return jnp.sin(y)
```

```python
jax.grad(foo)(3.)

> <ipython-input-113-b19a2dc1abf7>(12)debug_bwd()
-> return g
(Pdb) p x
Array(9., dtype=float32)
(Pdb) p g
Array(-0.91113025, dtype=float32)
(Pdb) q
```

+++

### `custom_vjp` functions can't be forward-differentiated

One important limitation: defining a `custom_vjp` rule makes the function
reverse-mode only. JAX has no way to derive a JVP rule from your VJP rules,
so applying forward-mode autodiff — `jax.jvp` or `jax.jacfwd` — to a
function that contains a `custom_vjp` function raises an error:

```{code-cell}
:tags: [raises-exception]

from jax import jvp

jvp(lambda x: clip_gradient(-1., 1., x), (3.,), (1.,))
```

Note that reverse mode *composes* fine: `jax.grad`-of-`grad`,
`jax.hessian`, and forward-over-reverse Hessian-vector products all work on
functions containing `custom_vjp`, because after one reverse-mode pass the
remaining computation is built from your `fwd` and `bwd` rules, which are
themselves differentiable. It's only forward mode applied to the
`custom_vjp` function directly that fails — as in `jax.jacfwd` of your
model, or `jax.jvp`-based sensitivity analysis. If you need both modes,
define a `custom_jvp` instead (JAX derives reverse mode from it
automatically), or use a hijax primitive ({doc}`custom-derivatives`), which
can carry rules for both modes at once.

+++

## More features and details

+++

### Working with `list` / `tuple` / `dict` containers (and other pytrees)

You should expect standard Python containers like lists, tuples, namedtuples, and dicts to just work, along with nested versions of those. In general, any {ref}`pytrees <jax-101-pytrees>` are permissible, so long as their structures are consistent according to the type constraints. 

Here's a contrived example with `jax.custom_jvp`:

```{code-cell}
from collections import namedtuple
Point = namedtuple("Point", ["x", "y"])

@custom_jvp
def f(pt):
  x, y = pt.x, pt.y
  return {'a': x ** 2,
          'b': (jnp.sin(x), jnp.cos(y))}

@f.defjvp
def f_jvp(primals, tangents):
  pt, = primals
  pt_dot, =  tangents
  ans = f(pt)
  ans_dot = {'a': 2 * pt.x * pt_dot.x,
             'b': (jnp.cos(pt.x) * pt_dot.x, -jnp.sin(pt.y) * pt_dot.y)}
  return ans, ans_dot

def fun(pt):
  dct = f(pt)
  return dct['a'] + dct['b'][0]
```

```{code-cell}
pt = Point(1., 2.)

print(f(pt))
```

```{code-cell}
print(grad(fun)(pt))
```

And an analogous contrived example with `jax.custom_vjp`:

```{code-cell}
@custom_vjp
def f(pt):
  x, y = pt.x, pt.y
  return {'a': x ** 2,
          'b': (jnp.sin(x), jnp.cos(y))}

def f_fwd(pt):
  return f(pt), pt

def f_bwd(pt, g):
  a_bar, (b0_bar, b1_bar) = g['a'], g['b']
  x_bar = 2 * pt.x * a_bar + jnp.cos(pt.x) * b0_bar
  y_bar = -jnp.sin(pt.y) * b1_bar
  return (Point(x_bar, y_bar),)

f.defvjp(f_fwd, f_bwd)

def fun(pt):
  dct = f(pt)
  return dct['a'] + dct['b'][0]
```

```{code-cell}
pt = Point(1., 2.)

print(f(pt))
```

```{code-cell}
print(grad(fun)(pt))
```

### Handling non-differentiable arguments

+++

Some use cases, like the final example problem, call for non-differentiable arguments like function-valued arguments to be passed to functions with custom differentiation rules, and for those arguments to also be passed to the rules themselves. In the case of `fixed_point`, the function argument `f` was such a non-differentiable argument. A similar situation arises with `jax.experimental.odeint`.

+++

#### `jax.custom_jvp` with `nondiff_argnums`

Use the optional `nondiff_argnums` parameter to `jax.custom_jvp` to indicate arguments like these. Here's an example with `jax.custom_jvp`:

```{code-cell}
from functools import partial

@partial(custom_jvp, nondiff_argnums=(0,))
def app(f, x):
  return f(x)

@app.defjvp
def app_jvp(f, primals, tangents):
  x, = primals
  x_dot, = tangents
  return f(x), 2. * x_dot
```

```{code-cell}
print(app(lambda x: x ** 3, 3.))
```

```{code-cell}
print(grad(app, 1)(lambda x: x ** 3, 3.))
```

Notice the gotcha here: no matter where in the argument list these parameters appear, they're placed at the *start* of the signature of the corresponding JVP rule. Here's another example:

```{code-cell}
@partial(custom_jvp, nondiff_argnums=(0, 2))
def app2(f, x, g):
  return f(g((x)))

@app2.defjvp
def app2_jvp(f, g, primals, tangents):
  x, = primals
  x_dot, = tangents
  return f(g(x)), 3. * x_dot
```

```{code-cell}
print(app2(lambda x: x ** 3, 3., lambda y: 5 * y))
```

```{code-cell}
print(grad(app2, 1)(lambda x: x ** 3, 3., lambda y: 5 * y))
```

#### `jax.custom_vjp` with `nondiff_argnums`

+++

A similar option exists for `jax.custom_vjp`, and, similarly, the convention is that the non-differentiable arguments are passed as the first arguments to the `_bwd` rule, no matter where they appear in the signature of the original function. The signature of the `_fwd` rule remains unchanged - it is the same as the signature of the primal function. Here's an example:

```{code-cell}
@partial(custom_vjp, nondiff_argnums=(0,))
def app(f, x):
  return f(x)

def app_fwd(f, x):
  return f(x), x

def app_bwd(f, x, g):
  return (5 * g,)

app.defvjp(app_fwd, app_bwd)
```

```{code-cell}
print(app(lambda x: x ** 2, 4.))
```

```{code-cell}
print(grad(app, 1)(lambda x: x ** 2, 4.))
```

See `fixed_point` above for another usage example.

**You don't need to use** `nondiff_argnums` **with array-valued arguments**, for example ones with integer dtype. Instead, `nondiff_argnums` should only be used for argument values that don't correspond to JAX types (essentially don't correspond to array types), like Python callables or strings. If JAX detects that an argument indicated by `nondiff_argnums` contains a JAX Tracer, then an error is raised. The `clip_gradient` function above is a good example of not using `nondiff_argnums` for integer-dtype array arguments.
