Skip to content
CalculatorBuddy

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.

Maintained by CalculatorBuddy TeamEditorial methodology

Add as a preferred source on Google (opens in a new tab)

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

Saved calculations stay only in this browser. Share copies a link containing your entered values.

Function and derivative

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

Compare scenarios

Keep a baseline, then change an assumption to see its effect side by side.

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.

How to use this calculator

Enter the function and the point where you want its slope. Check that nearby values are defined. A positive derivative means the function is increasing locally, while a negative derivative means it is decreasing; a numerical estimate alone cannot prove differentiability.

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.

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).

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

Numerical and symbolic limits

The derivative describes local rate of change where the function is differentiable. Domain boundaries, corners, discontinuities, and piecewise definitions can make a derivative one-sided or undefined. Simplified output should be checked against the original domain, because algebraic cancellation can hide excluded points.

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.

Related calculators