Calculate the derivative of a function quotient and see the step-by-step breakdown.
Input Expressions
Resulting Derivative
[(x)(cos(x)) - (sin(x))(1)] / (x)²
Result (d/dx [f/g])
Application of the quotient rule formula.
Formula Breakdown
Low · dHigh(x) · (cos(x))
High · dLow(sin(x)) · (1)
Denominator²(x)²
About this calculator
Quick Reference
Numerator fHigh
Denominator gLow
Resulting SignSubtraction (-)
What is the Quotient Rule?
The quotient rule is a formal rule used in calculus to find the derivative of a function that is the ratio of two other functions. If y = f(x) / g(x), then its derivative is given by the difference of g(x)f'(x) and f(x)g'(x), all divided by the square of g(x).
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Pro Tips
Order Matters: Unlike the product rule, the subtraction in the numerator means you must do g·f' first.
Denominator Square: Never forget to square the original denominator in the result.
Rhyme: 'Low d-High minus High d-Low, square the bottom and off you go!'
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Fun Facts
"The quotient rule can actually be derived using the product rule and the chain rule by writing f(x)/g(x) as f(x) · [g(x)]⁻¹."
"Gottfried Wilhelm Leibniz, the co-inventor of calculus, developed many of these foundational rules."
