Quotient Rule Calculator

The quotient rule differentiates a fraction of two functions. Given f(x) = u(x)/v(x), the derivative is:

f’(x) = [u’(x)v(x) - u(x)v’(x)] / [v(x)]^2

The order matters here: numerator derivative times denominator, minus numerator times denominator derivative, all divided by denominator squared. Students often reverse the subtraction. A handy phrase: “low d-high minus high d-low, over low squared.” The d means “derivative of.”

This calculator uses u(x) = ax^m as the numerator, and one of five denominator types: a monomial bx^n, sin(bx), cos(bx), the natural exponential, or the natural log.

For each v(x), the derivative v’(x) is:

• Monomial bx^n: nbx^(n-1)
• sin(bx): b·cos(bx)
• cos(bx): -b·sin(bx)
• e^x: e^x
• ln(x): 1/x (requires x > 0)

The result is undefined wherever v(x) = 0, since that makes the denominator of the original fraction zero (and the denominator of the derivative, v², also zero).

A worked example: f(x) = 3x²/sin(2x). At x = 1: u = 3, u’ = 6, v = sin(2) ≈ 0.909, v’ = 2cos(2) ≈ -0.832. Quotient rule: [6(0.909) - 3(-0.832)] / (0.909)² = [5.454 + 2.496] / 0.826 ≈ 9.62.

The quotient rule can always be replaced by the product rule with a negative exponent: d/dx[u/v] = d/dx[u·v^(-1)] and then applying the chain rule to v^(-1). Many mathematicians prefer this approach to avoid memorizing a separate rule, but both paths give the same answer.