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

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

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 $g(x)$:

2. Compute the derivatives:

3. Apply the product rule:

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)