Derivative Calculator – Instant Step-by-Step Solutions
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How to Use This Derivative Calculator

Using this derivative calculator is simple. Type your function in the input box using standard math notation — for example, 3*x^2 + sin(x) — then press Calculate Derivative. The calculator instantly displays the derivative and full step-by-step working so you can learn how each rule was applied.

Supported Syntax

Use * for multiplication, ^ for exponents, and parentheses for grouping. Functions like sin(x), cos(x), tan(x), ln(x), e^x, and sqrt(x) are all supported. For higher-order derivatives, change the order value — for example, set it to 2 for the second derivative.

Core Differentiation Rules

Every derivative is computed using one or more of these fundamental rules of calculus:

Power Rule
d/dx [xⁿ] = n·xⁿ⁻¹
Constant Rule
d/dx [c] = 0
Sum Rule
d/dx [f+g] = f'+g'
Product Rule
d/dx [fg] = f'g + fg'
Quotient Rule
d/dx [f/g] = (f'g–fg')/g²
Chain Rule
d/dx [f(g)] = f'(g)·g'
Sine
d/dx [sin x] = cos x
Cosine
d/dx [cos x] = –sin x
Tangent
d/dx [tan x] = sec²x
Exponential
d/dx [eˣ] = eˣ
Natural Log
d/dx [ln x] = 1/x
Square Root
d/dx [√x] = 1/(2√x)

Frequently Asked Questions

What is a derivative in calculus?

A derivative measures how a function changes as its input changes. Geometrically, it represents the slope of the tangent line to the function's graph at any given point. Derivatives are fundamental to calculus and are used in physics, engineering, economics, and many other fields to model rates of change.

How do I find the derivative of a polynomial?

Use the power rule: multiply each coefficient by its exponent, then reduce the exponent by 1. For example, the derivative of 4x³ is 12x². Apply this to every term and use the sum rule to combine — the derivative of 4x³ + 2x − 7 is 12x² + 2.

What is a second derivative?

The second derivative is the derivative of the derivative. It tells you how the rate of change itself is changing — essentially the concavity or curvature of a function. In physics, if position is f(t), the first derivative is velocity and the second derivative is acceleration. Set the order to 2 in this calculator to compute it instantly.

When should I use the chain rule?

Use the chain rule when differentiating a composite function — a function inside another function. For example, sin(x²) requires the chain rule: differentiate the outer function (cos) and multiply by the derivative of the inner function (2x), giving 2x·cos(x²).

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