CalcLibrary

Derivative Calculator

Find the first, second, or third symbolic derivative of an expression with respect to a variable.

Examples

Polynomial

Differentiate x^3 + 2x^2 - 5x + 3 and compare all three orders.

Expression
x^3 + 2x^2 - 5x + 3
Variable
x
Selected derivative
3 * x ^ 2 + 4 * x - 5
First derivative
3 * x ^ 2 + 4 * x - 5
Second derivative
6 * x + 4
Third derivative
6
Original function
x^3 + 2x^2 - 5x + 3
Order
First derivative
Operation
d/dx

Examples

How It Works

Formula

f′(x)=ddxf(x)f\prime(x)=\frac{d}{dx}f(x)

f′′(x)=d2dx2f(x)f\prime\prime(x)=\frac{d^2}{dx^2}f(x)

f′′′(x)=d3dx3f(x)f\prime\prime\prime(x)=\frac{d^3}{dx^3}f(x)

Variables

f(x)f(x)

Original function

xx

Variable of differentiation

f′(x)f\prime(x)

First derivative: slope

f′′(x)f\prime\prime(x)

Second derivative: curvature

f′′′(x)f\prime\prime\prime(x)

Third derivative: change in curvature

The expression is parsed by mathjs, then differentiated once, twice, and three times. Each derivative is simplified, and the selected order is shown as a readable formula.

Frequently Asked Questions

01What does a derivative mean?
A derivative tells you how an expression changes as the variable changes. The first derivative is the rate of change of the original expression.
02What is the first derivative?
The first derivative shows increase or decrease. Positive values mean the function is rising, negative values mean it is falling.
03What is the second derivative?
The second derivative shows how the first derivative is changing. It helps you see whether the rate of change is increasing or decreasing.
04What is the third derivative?
The third derivative shows how the second derivative changes. It is useful in calculus and motion problems, but beginners can treat it as one more change-of-change layer.
05What syntax should I use?
Use x^2 for powers, 2*x for multiplication when needed, and parentheses around function inputs, such as sin(x) or sqrt(x).
06Are sine, cosine, and square roots supported?
Yes. Try sin(x), cos(x), tan(x), exp(x), log(x), and sqrt(x). Use the same variable name in the expression and variable box.
07How do I choose the variable?
Use the variable that the expression changes with, such as x or t. If the expression has more than one symbol, the calculator treats the others as constants while differentiating with respect to the variable you enter.
08What input mistakes are common?
Use ^ for powers, include closing parentheses, and avoid spaces inside the variable name. For example, write x^3 + 2*x, not x3 + 2x if parsing fails.

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