About the Quotient Rule Calculator
Many functions in calculus are fractions: rational functions such as (x² + 1)/(x − 3), trigonometric ratios such as sin(x)/x, and growth models with an exponential on top or bottom. To differentiate a fraction you need the quotient rule, and it is easy to slip on the order of the terms or forget to square the denominator.
This quotient rule calculator takes the numerator u(x) and the denominator v(x) as you would write them, differentiates each one symbolically, substitutes them into the quotient rule and shows the result. If you give a point, it also evaluates the slope there and writes the equation of the tangent line, so you can check homework answers or graphing work in one place.
How to Use the Quotient Rule Calculator
Enter the numerator u(x), the expression above the fraction bar. Use x as the variable, ^ for powers, and functions such as sin(x), cos(x), tan(x), ln(x), log(x), sqrt(x), exp(x) or e^x. Implicit multiplication like 2x or 3sin(x) is understood.
Enter the denominator v(x), the expression below the fraction bar.
Optionally enter a point to evaluate the derivative there. It can be a number such as 2 or −0.5, or an expression such as pi/4.
The Formulas
f(x) = u(x) / v(x)
f′(x) = (u′(x) · v(x) − u(x) · v′(x)) / v(x)²
tangent at x = a: y = f(a) + f′(a)(x − a)Step-by-Step Example
Differentiate f(x) = (x² + 1)/(x − 3) and find the tangent line at x = 1.
u = x² + 1 u′ = 2x
v = x − 3 v′ = 1
f′(x) = (2x(x − 3) − (x² + 1) · 1) / (x − 3)²
at x = 1: f′(1) = (2 · (−2) − 2) / (−2)² = −6 / 4 = −1.5
f(1) = 2 / (−2) = −1
tangent: y = −1 − 1.5(x − 1) = −1.5x + 0.5Expanding the numerator gives 2x² − 6x − x² − 1 = x² − 6x − 1, so a textbook might write the answer as (x² − 6x − 1)/(x − 3)². Both forms are equal for every x except 3, where the function itself is undefined.
Why the Order Matters
The product rule is symmetric: (uv)′ = u′v + uv′, and swapping the terms changes nothing. The quotient rule has a subtraction, so the order is fixed. Writing uv′ − u′v instead of u′v − uv′ flips the sign of the whole answer, which is one of the most common errors in first-year calculus. The rhyme "low d-high minus high d-low, over the square of what's below" keeps the order straight: the denominator times the derivative of the numerator comes first.
A quick check is to test a simple case you already know. For f(x) = 1/x, the quotient rule gives (0 · x − 1 · 1)/x² = −1/x², which matches the power rule applied to x⁻¹. If your version of the rule gives +1/x², the order is reversed.
Quotient Rule Versus Other Methods
Any quotient can be rewritten as a product, u · v⁻¹, and differentiated with the product rule and the chain rule. That route gives u′v⁻¹ − uv′v⁻², which simplifies to the same quotient rule result. Some students prefer it because there is one less formula to remember; others find the quotient rule tidier for rational functions.
When the denominator is a constant, the quotient rule is unnecessary: (x² + 1)/5 is just (1/5)(x² + 1), whose derivative is 2x/5. When the numerator is a constant, as in 3/(x² + 1), rewriting as 3(x² + 1)⁻¹ and using the chain rule is often quicker, giving −6x/(x² + 1)². The calculator handles all of these cases, so you can compare the forms.
Supported Functions and Input Tips
The calculator differentiates powers, polynomials, sin, cos, tan, sec, csc and cot, inverse trigonometric functions (asin, acos, atan), hyperbolic functions (sinh, cosh, tanh), sqrt, natural log ln, base-10 log, exp and absolute value, with the chain rule applied automatically. Powers such as x^x, where both the base and the exponent contain x, are handled through logarithmic differentiation.
Use brackets generously. Write sin(2x) rather than sin 2x, and e^(2x) rather than e^2x, which means e² · x. Write 1/(x + 1) for a fraction with a sum underneath; 1/x + 1 means one over x, plus one. Angles in trigonometric functions are in radians, as in all calculus.
Where Quotient Rule Derivatives Are Used
The derivative of a quotient appears whenever a rate is a ratio. Average cost per unit is total cost divided by quantity, and its derivative shows whether making more units lowers the average cost. Concentration in a mixing tank is the amount of substance divided by the volume, both changing in time. The tangent function itself is sin/cos, and the quotient rule is how its derivative sec²(x) is first derived.
Finding where a quotient's derivative is zero locates maximums and minimums. For f(x) = x/(x² + 1), the derivative (x² + 1 − 2x²)/(x² + 1)² is zero when x² = 1, so the function peaks at x = 1 with value 0.5 and bottoms out at x = −1.
Understanding Your Result
The headline shows f′(x) as a formula. It is tidied, so terms like 1 · v are simplified, but brackets are not expanded.
The u, v and their derivatives line lists the four pieces that go into the rule.
The substituted line shows the rule with those pieces in place, before simplification, matching how you would write it by hand.
The at the point line gives the slope f′(a) and the function value f(a).
The tangent line gives the line touching the curve at that point in the form y = mx + b.
When Should You Use This Calculator?
Use it to check a derivative you worked out by hand.
Use it to find the slope or tangent line of a rational or trigonometric function at a point.
Use it to see the intermediate steps when learning the quotient rule.
Common Mistakes
Reversing the subtraction. It is u′v − uv′, not uv′ − u′v.
Forgetting to square the denominator. The bottom of the answer is v², not v.
Missing brackets in the input. Write e^(2x) and 1/(x + 1), not e^2x and 1/x + 1.
Evaluating where v = 0. The function and its derivative are undefined there.