Quotient Rule Calculator. Enter your numerator function f(x) and denominator function g(x) to apply the Quotient Rule and compute the derivative of f(x)/g(x). The calculator returns f'(x), the full derivative expression, and shows the step-by-step formula breakdown: [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]². Supports polynomials, trig functions, exponentials, and more. Also try the Arc Length Calculator (Calculus).
The Quotient Rule is a differentiation formula used when you need to find the derivative of one function divided by another. If h(x) = f(x)/g(x), then h'(x) = [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]². It is one of the fundamental rules of differential calculus alongside the Product Rule and Chain Rule. See also our Critical Points Calculator.
Use the Quotient Rule when the fraction cannot be easily simplified into a product or simpler terms. For example, (x² + 1)/(x + 3) is best differentiated directly with the Quotient Rule. However, something like x²/x can be simplified to x first, making the derivative trivial.
The formula is: d/dx [f(x)/g(x)] = [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]². A helpful mnemonic is 'low d-high minus high d-low, all over low squared' — where 'high' is the numerator and 'low' is the denominator.
Yes. You can write f(x)/g(x) as f(x)·[g(x)]⁻¹ and apply the Product Rule combined with the Chain Rule on [g(x)]⁻¹. The result is algebraically equivalent to the standard Quotient Rule formula.
If g(x) = 0 at a specific point, the original function f(x)/g(x) is undefined there, and so is its derivative. The Quotient Rule is only valid at points where g(x) ≠ 0. Always check the domain of your function before differentiating.
Yes, absolutely. The numerator is f'(x)·g(x) minus f(x)·g'(x) — not the other way around. Reversing the order gives the wrong sign and an incorrect derivative. This is one of the most common mistakes students make.
Yes. For example, tan(x) = sin(x)/cos(x), and applying the Quotient Rule yields sec²(x). Similarly, cot(x), sec(x), and csc(x) can all be derived using the Quotient Rule from their sine and cosine definitions.
This calculator symbolically identifies common function types including polynomials, exponentials (e^x), sine, cosine, and natural logarithm. It computes their derivatives analytically and applies the Quotient Rule formula to produce the result at a given point x = 1 for numerical preview. You might also find our dy/dx (Derivative Expression) — Implicit Differentiation useful.