differentiate the function calculator

Enter a function of x, choose derivative order, and optionally evaluate the derivative at a specific point.

Function f(x)

Derivative order (n)

Evaluate at x = (optional)

Decimal places for numeric output

Try: polynomial · product rule · chain rule · quotient form

How to use this calculator

This tool computes symbolic derivatives (exact formulas) and can also evaluate the derivative at a point. It is useful for calculus homework, quick verification, and understanding how functions change.

Step 1: Enter a function in terms of x.
Step 2: Choose derivative order (1 for first derivative, 2 for second derivative, etc.).
Step 3: Optionally enter an x-value to get a numeric slope value.
Step 4: Click Differentiate.

Input format and supported syntax

Recommended function syntax

Use ^ for powers: x^5
Use explicit multiplication: 3*x, x*sin(x)
Use parentheses for clarity: (x^2+1)/(x-3)
You can use ln(...) or log(...) for natural log
Common functions: sin, cos, tan, exp, sqrt, abs

Examples you can paste

x^3 + 2*x - 9
sin(x)^2 + cos(x)^2
e^x * ln(x+1)
sqrt(x^2+4)

Why derivatives matter

Derivatives measure rate of change. In geometry, the derivative gives slope of the tangent line. In physics, derivatives describe velocity and acceleration. In economics, derivatives model marginal cost and marginal revenue. In machine learning, gradients are derivatives that drive optimization.

Core differentiation rules (quick reference)

Power rule

If f(x)=x^n, then f'(x)=n*x^(n-1).

Linearity

The derivative of a sum is the sum of derivatives, and constants factor out.

Product rule

If f(x)=u(x)v(x), then f'(x)=u'(x)v(x)+u(x)v'(x).

Quotient rule

If f(x)=u(x)/v(x), then f'(x)=(u'v-uv')/v^2.

Chain rule

If f(x)=g(h(x)), then f'(x)=g'(h(x))*h'(x). This is essential for nested functions like sin(x^2) or ln(3x+1).

Tips for better results

Use parentheses generously.
If an expression fails, rewrite it with explicit multiplication.
Check domain restrictions (for example, ln(x) requires x > 0).
Use higher derivative orders to inspect curvature and inflection behavior.

Final note

This calculator is designed to be practical and fast for common symbolic differentiation tasks. It is a great companion for algebra and calculus work when you want both an exact derivative formula and a numerical derivative value at a specific point.