Product rule

The product rule is a fundamental concept in calculus used to compute the derivative of the product of two functions. If you have two functions, $f(x)$ and $g(x)$, the product rule helps you find the derivative of their product

$$h(x) = f(x)g(x)$$

with respect to $x$. Mathematically, the product rule states that:

$$h'(x) = f'(x)g(x) + f(x)g'(x)$$

where $h'(x)$ is the derivative of $h(x)$ with respect to $x$, $f'(x)$ is the derivative of $f(x)$ with respect to $x$, and $g'(x)$ is the derivative of $g(x)$ with respect to $x$.

Examples

Find the derivative of the product function $h(x) = (x^2 + 1)(3x - 4)$ with respect to $x$

1. Define the functions $f(x)$ and $g(x)$:

$$\begin{aligned} f(x) &= x^2 + 1 \\ g(x) &= 3x - 4 \end{aligned}$$

2. Compute the derivatives:

$$\begin{aligned} f'(x) &= 2x \\ g'(x) &= 3 \end{aligned}$$

3. Apply the product rule:

$$\begin{aligned} h'(x) &= f'(x)g(x) + f(x)g'(x) \\ &= 2x(3x - 4) + (x^2 + 1)(3) \\ &= 9x^2 - 8x + 3 \\ \end{aligned}$$

In Sympy this looks like:

import sympy

x = sympy.symbols("x")

u_x = x**2 + 1
v_x = 3 * x - 4

u_prime = sympy.diff(u_x, x)
v_prime = sympy.diff(v_x, x)

w_prime = u_prime * v_x + u_x * v_prime

sympy.simplify(w_prime)