Derivative calculator with steps
Type a function of x and get its derivative in simplest form, with the rules used at each step: power, product, quotient, chain and the standard derivatives of exponentials, logarithms and trig functions. One click graphs the function and its derivative together.
- Product rule: (f·g)′ = f′·g + f·g′ with f = x², g = sin(x)
- f′ = 2x, g′ = cos(x)
- Power rule: d/dx[x²] = 2 · (x)^(2 − 1)
- d/dx[sin(x)] = cos(x)
- Result: 2x * sin(x) + x²cos(x)
How to use it
- 1Type a function of x the way you’d write it:
x^2 sin(x),e^(3x)/x,ln(cos(x)). - 2Press Differentiate. The result is simplified, and the steps list the rules used.
- 3Use the link under the result to graph the function and its derivative and see where the slope is zero.
- 4In the full calculator you can also type
f'(x)ord/dx(x^3)directly in a row.
The rules it uses
| Rule | Statement |
|---|---|
| Power | ddx(xn)=nxn−1 |
| Product | (fg)′ = f′g + fg′ |
| Quotient | (f/g)′ = (f′g − fg′)/g² |
| Chain | (f(g(x)))′ = f′(g(x)) · g′(x) |
| Exponential | (eˣ)′ = eˣ, (aˣ)′ = aˣ ln a |
| Logarithm | (ln x)′ = 1/x |
| Trigonometric | (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = sec²x |
How it works
The expression is parsed into a tree and differentiated symbolically, rule by rule, so the result is exact rather than a numerical approximation. A computer algebra system then simplifies it. The steps are generated from the same tree: each node of the expression contributes the rule that applies to it.
Simplification is a matter of taste, so the answer may look different from your textbook’s while being equal to it. Graphing both is a quick way to check: identical curves mean identical functions.
What derivatives tell you
- Where the derivative is zero, the function has a horizontal tangent: a maximum, a minimum or a stationary inflection point.
- Where it is positive the function is increasing; where negative, decreasing.
- The second derivative tells you which way the graph bends: positive means concave up.
Frequently asked questions
Does the derivative calculator show steps?
How do I enter a function?
Can it find second derivatives?
Is the answer exact?
Related tools and guides
Antiderivatives and definite integrals, exact where possible.
Direction fields for dy/dx = f(x, y), with solutions through any point.
Linear, polynomial, exponential or any custom model, with error bars.
Surfaces, implicit and parametric 3D graphs, exportable for 3D printing.
Type f′(x) next to any function and the derivative is graphed live as you change it.