The Quotient Rule Calculator is intended for derivatives of functions written as one expression divided by another. The quotient rule avoids the need to rewrite every quotient as a product and is especially convenient when both numerator and denominator vary with x.
Quotient Rule
If y = f(x)/g(x), then y′ = [g(x)f′(x) − f(x)g′(x)]/[g(x)]². The denominator is squared, while the numerator follows the order “denominator times derivative of numerator minus numerator times derivative of denominator.”
How to Use It
Enter the numerator and denominator exactly as functions. Use parentheses around grouped expressions so that powers and signs are interpreted correctly. Calculate the derivative and simplify the result if a cleaner algebraic form is useful.
Example
For y = x²/(x+1), let f=x² and g=x+1. Then f′=2x and g′=1. The quotient rule gives [(x+1)(2x) − x²]/(x+1)², which simplifies to (x²+2x)/(x+1)².
Domain Matters
The original denominator cannot be zero. Even if a simplified derivative looks harmless at a particular value, the original function still has restrictions that should be respected when interpreting the result.
Quick Verification
For a quotient, carefully check the minus sign between the two numerator terms. That sign is one of the most common sources of incorrect manual derivatives.