Quotient Rule

import sympy

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 \(x\).

Example

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 \(g(x)\):

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

  1. Compute the derivatives:

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

  1. 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:

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)

\(\displaystyle \frac{3 x^{2} - 8 x - 3}{9 x^{2} - 24 x + 16}\)