quickstart tweaks (from jax-ml#20819)

docs/tutorials/quickstart.md

@@ -14,7 +14,7 @@ kernelspec:
 
 # Quickstart
 
-**JAX a library for array-oriented numerical computation (*a la* [NumPy](https://numpy.org/)), with automatic differentiation and JIT compilation to enable high-performance machine learning research**.
+**JAX a library for array-oriented numerical computation (*à la* [NumPy](https://numpy.org/)), with automatic differentiation and JIT compilation to enable high-performance machine learning research**.
 
 This document provides a quick overview of essential JAX features, so you can get started with JAX quickly:
 
@@ -125,16 +125,18 @@ In the above example we jitted `sum_logistic` and then took its derivative. We c
 print(grad(jit(grad(jit(grad(sum_logistic)))))(1.0))
 ```
 
-Similarly, the {func}`jax.jacobian` transformation can be used to compute gradients of vector-valued functions:
+Beyond scalar-valued functions, the {func}`jax.jacobian` transformation can be
+used to compute the full Jacobian matrix for vector-valued functions:
 
 ```{code-cell}
 from jax import jacobian
 print(jacobian(jnp.exp)(x_small))
 ```
 
-For more advanced autodiff operations, you can use {func}`jax..jacrev` for reverse-mode vector-Jacobian products,
-and {func}`jax.jacfwd` for forward-mode Jacobian-vector products.
+For more advanced autodiff operations, you can use {func}`jax.vjp` for reverse-mode vector-Jacobian products,
+and {func}`jax.jvp` and {func}`jax.linearize` for forward-mode Jacobian-vector products.
 The two can be composed arbitrarily with one another, and with other JAX transformations.
+For example, {func}`jax.jvp` and {func}`jax.vjp` are used to define the forward-mode {func}`jax.jacfwd` and reverse-mode {func}`jax.jacrev` for computing Jacobians in forward- and reverse-mode, respectively.
 Here's one way to compose them to make a function that efficiently computes full Hessian matrices:
 
 ```{code-cell}