Quotient Rule Calculator
Enter the top and bottom functions to differentiate the fraction using the quotient rule.
Use ^ for powers and x as the variable, e.g. sin(x), x^2.
The quotient rule differentiates a fraction of two functions. If y = u/v, then y′ = (u′v − uv′) ÷ v², where u′ and v′ are the derivatives of the top and bottom. The derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all over the denominator squared.
How to Use This Calculator
Enter the numerator and denominator as functions of x, using ^ for powers and standard function names like sin, cos and ln. The calculator differentiates the fraction u/v and shows the derivatives of the top and bottom along the way.
The quotient rule is the rule for differentiating any fraction where both the top and bottom depend on the variable, and it appears constantly in calculus.
How the Calculation Works
The quotient rule takes the derivative of the numerator times the denominator, subtracts the numerator times the derivative of the denominator, and divides the whole thing by the denominator squared. The order of the subtraction matters — swapping the two terms flips the sign and gives the wrong answer.
The calculator computes u′ and v′ separately, substitutes them into the formula, and simplifies the result.
(u/v)′ = (u′·v − u·v′) ÷ v²
Worked Example
Differentiate x² ÷ (x + 1). Here u = x², so u′ = 2x; and v = x + 1, so v′ = 1.
Apply the rule: (2x·(x + 1) − x²·1) ÷ (x + 1)² = (2x² + 2x − x²) ÷ (x + 1)² = (x² + 2x) ÷ (x + 1)². The calculator shows this in the equivalent factored form x(x + 2)(x + 1)⁻².
Assumptions and Limitations
- Differentiates with respect to x
- Enter standard functions (sin, cos, ln, exp) and ^ for powers
- The result is shown in an equivalent, possibly reordered, form
- For a product rather than a quotient, use the product rule
When to Use a Quotient Rule Calculator
Calculus students use this to differentiate a fraction of two functions with the quotient rule. Run it to check a derivative, follow the rule step by step, or handle a messy quotient quickly. Enter the numerator and denominator to get the derivative.
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Open calculator →Frequently Asked Questions
What is the quotient rule?
It is the rule for differentiating a fraction of two functions: (u/v)′ = (u′v − uv′) ÷ v². You differentiate the top times the bottom, minus the top times the derivative of the bottom, over the bottom squared.
Does the order of subtraction matter?
Yes. It must be u′v − uv′, not the other way around. Reversing the two terms negates the answer, which is one of the most common quotient-rule mistakes.
When do I use the quotient rule instead of the product rule?
Use the quotient rule when one function is divided by another. Use the product rule when they are multiplied. A quotient can also be rewritten as a product with a negative power if you prefer.
How do I enter functions like sin or ln?
Type them by name with the argument in parentheses, such as sin(x) or ln(x). Use ^ for powers, so x squared is x^2, and x is the variable of differentiation.
Why does my answer look different from the textbook?
Equivalent derivatives can be written in several forms. The calculator may expand or order terms differently, but the result is mathematically the same as a correctly simplified textbook answer.
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