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Derivative Calculator

Differentiate a function of x through the tenth derivative, evaluate it at a point, find the tangent line, and compare function and derivative graphs.

ExampleSample values — edit any field to see your result.

Supports powers, roots, exp/log, trigonometric functions, e, and pi.

Derivative

2+3*x^2

Estimated result

Function and derivative

-4-2024-10-50510xyy = 2*x+x^3y = 2+3*x^2

Successive derivative steps

StepExpression
f(x)2*x+x^3
f′(x)2+3*x^2
Derivative2+3*x^2

Differentiate a function of x symbolically, simplify the result, find a first through tenth derivative, and optionally evaluate it at a point. A graph compares the original function with its derivative and the steps show each derivative order.

Formula

The derivative is the limit of the difference quotient:

f′(x) = lim[h→0] (f(x+h) − f(x)) / h

At x = a, the tangent line is y = f(a) + f′(a)(x − a).

Examples

Differentiate x³ − 4x

The first derivative is 3x² − 4. At x = 2, the slope is 8, so the tangent line passes through f(2) = 0 with slope 8.

Second derivative

Choosing order 2 differentiates the first result again. For x³ − 4x, the second derivative is 6x.

Differentiate a simplified expression, but keep parentheses explicit whenever the numerator or denominator has multiple terms.

Frequently asked questions

What does the derivative represent?
The derivative is the instantaneous rate of change of a function and the slope of its tangent line.
Can I calculate higher derivatives?
Yes. Choose an order from one through ten to differentiate repeatedly with respect to x.
Can the calculator evaluate the derivative at a point?
Yes. Enter an x-value to see the function value, derivative value, and tangent-line equation there.
What notation should I use?
Use x as the variable, ^ for powers, parentheses, and functions such as sin, cos, exp, ln, and sqrt.

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