Derivative Calculator
Differentiate any function and see which rules were used — power, product, quotient and chain rule.
Press Enter to solve · ^ for powers · sqrt() for roots · separate a system with a comma
The derivative calculator differentiates a function with respect to x and explains the rules it applied. Type x^3 sin(x), e^(2x)/x or ln(x^2+1); for a second derivative type second derivative of x^4.
How to use the derivative calculator
- Enter the function f(x). You can also start with
d/dx. - Press Enter.
- See each term differentiated, the rules used and a graph comparing f(x) with f′(x).
Derivative rules
| Rule | Formula |
|---|---|
| Power rule | d/dx xⁿ = n·xⁿ⁻¹ |
| Product rule | (uv)′ = u′v + uv′ |
| Quotient rule | (u/v)′ = (u′v − uv′) / v² |
| Chain rule | d/dx f(g(x)) = f′(g(x))·g′(x) |
| Trig | (sin x)′ = cos x, (cos x)′ = −sin x |
| Exponential & log | (eˣ)′ = eˣ, (ln x)′ = 1/x |
Example: derivative of ln(x² + 1)
- Outer function ln(u) has derivative 1/u.
- Inner function u = x² + 1 has derivative 2x.
- Chain rule: f′(x) = 2x / (x² + 1).
Questions about derivatives
What does a derivative tell you?
The derivative f′(x) gives the instantaneous rate of change of f, which is the slope of the tangent line to the graph at each point.
How do I know when to use the chain rule?
Use the chain rule whenever one function is inside another, such as sin(3x), (2x + 1)⁵ or e^(x²). The calculator lists it among the rules used when it applies.
Can it find partial derivatives?
Add “with respect to y” to differentiate by another variable, for example derivative of x^2*y^3 with respect to y; other letters are treated as constants.