Math

Differentiation Calculator

Symbolic Differentiation with Step-by-Step Solutions

Differentiation is the process of finding the derivative of a function. This calculator performs symbolic differentiation on any mathematical expression, showing each rule applied along the way. Whether you call it "differentiation" or "finding the derivative," the result is the same: the rate of change of your function.

Derivative Calculator Tips

Click to show tips

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Power Rule

Differentiate a polynomial using the power rule.

Key values: f(x) = x^3 - 2x^2 + x · 1st derivative

Chain Rule

Differentiate a composite function with sin(x^2).

Key values: f(x) = sin(x^2) · Chain rule applied

Product Rule with Tangent

Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.

Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0

Second Derivative

Compute the second derivative to analyze concavity.

Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative

Documentation

This calculator is also known as Differentiation Calculator.

Read the complete guide

What Is Differentiation?

Differentiation is the mathematical operation that computes the derivative of a function. Given a function f(x), differentiation produces f'(x), which measures how f changes as x changes.

Key Differentiation Rules

These fundamental rules cover most differentiation problems:

Power Rule: d/dx[x^n] = n*x^(n-1)
Product Rule: d/dx[f*g] = f'g + fg'
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Quotient Rule: d/dx[f/g] = (f'g - fg') / g^2

Examples

Polynomial Differentiation

Differentiate f(x) = 3x^4 - 2x^2 + 5.

Apply the power rule to each term: d/dx[3x^4] = 12x^3, d/dx[-2x^2] = -4x, d/dx[5] = 0.

Key takeaway: The power rule handles each polynomial term independently.

Chain Rule Application

Differentiate f(x) = sin(x^2).

The outer function is sin(u), the inner is u = x^2. By the chain rule: cos(x^2) * 2x = 2x*cos(x^2).

Key takeaway: Composite functions always require the chain rule.

Getting Better at Differentiation

Practice these strategies to improve your differentiation skills:

Memorize the basic derivative rules before attempting complex problems
Always identify the outermost operation first to determine which rule applies
Simplify expressions before differentiating when possible
Check your answer by plotting both the function and its derivative

Frequently Asked Questions about Differentiation Calculator

Is differentiation the same as finding the derivative?

Yes, differentiation is the process, and the derivative is the result. "Differentiate f(x)" and "find the derivative of f(x)" mean exactly the same thing.

What is the difference between d/dx and partial d/dx?

d/dx denotes the ordinary derivative for single-variable functions. The partial derivative symbol is used when the function depends on multiple variables and you differentiate with respect to one while holding the others constant.