Quotient rule

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

$$h(x) = \frac{f(x)}{g(x}$$

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

$$h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{f(x)^2}$$

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

Example: Find the derivative of the quotient function $h(x) = \frac{x^2 + 1}{3x - 4}$ 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 quotient rule:

$$\begin{aligned} h'(x) &= \frac{f'(x)g(x) - f(x)g'(x)}{f(x)^2} \\ &= \frac{2x(3x - 4) - 3(x^2 + 1)}{(3x - 4)^2} \\ &= \frac{3x^2 - 8x - 3}{9x^2 - 24x + 16} \end{aligned}$$

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) / v_x**2

sympy.simplify(w_prime)