What is a derivative in calculus?

A derivative measures the instantaneous rate of change of a function at any given point. If f(x) represents position, then f'(x) represents velocity — how fast position changes. Geometrically, the derivative at a point equals the slope of the tangent line to the curve at that point.

What is the Power Rule for derivatives?

The Power Rule states: d/dx(xⁿ) = n·xⁿ⁻¹. Bring the exponent down as a coefficient, then reduce the exponent by 1. Example: d/dx(x⁴) = 4x³. For a constant multiple: d/dx(3x²) = 6x.

What is the Chain Rule?

The Chain Rule differentiates composite functions: d/dx[f(g(x))] = f'(g(x)) · g'(x). Example: d/dx(sin(x²)) = cos(x²) · 2x = 2x·cos(x²). Always multiply by the derivative of the inner function.

What is the Product Rule?

The Product Rule: d/dx(u·v) = u'·v + u·v'. Example: d/dx(x²·sin(x)) = 2x·sin(x) + x²·cos(x). Use when differentiating two functions multiplied together.

What is the Quotient Rule?

The Quotient Rule: d/dx(u/v) = (u'·v − u·v') / v². Example: d/dx(x²/sin(x)) = (2x·sin(x) − x²·cos(x)) / sin²(x). Remember: 'low d-high minus high d-low, square the bottom and away we go.'

What are the derivatives of trigonometric functions?

Key trig derivatives: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(sec x) = sec x · tan x, d/dx(csc x) = −csc x · cot x, d/dx(cot x) = −csc²x. These are multiplied by the inner derivative when using Chain Rule.

What is the derivative of e^x and ln(x)?

d/dx(eˣ) = eˣ (the exponential function is its own derivative!). d/dx(ln x) = 1/x. With Chain Rule: d/dx(e^(2x)) = 2e^(2x), and d/dx(ln(x²)) = 2x/x² = 2/x.

What is the second derivative?

The second derivative f''(x) is the derivative of the first derivative f'(x). It measures the rate of change of the rate of change — i.e., acceleration if f(x) is position. It also tells you concavity: f''(x) > 0 means concave up, f''(x) < 0 means concave down.

Derivative Calculator

Our Derivative Calculator is a powerful symbolic differentiation tool that computes exact derivatives with detailed step-by-step explanations. Enter any mathematical function and get the derivative immediately, with each differentiation rule clearly shown. Supports all standard calculus functions including polynomials, sine, cosine, tangent, exponentials, natural logarithms, and more. Perfect for calculus students, engineers, and anyone needing to differentiate functions quickly.

star 4.9 (850 ratings)

calendar_today Updated: July 2025

auto_awesome AI Insights AI

New

Derivative Order

Function f(x)

Use ^ for powers, * for multiplication

Variable

Quick Examples

Evaluate at a point

functions Result

f'(x) =

3x² + 4x − 5

format_list_numbered Step-by-Step Solution

Rules Applied

Input Values

Function f(x)

Variable

Derivative Order

1st Derivative f'(x)

2nd Derivative f''(x)

3rd Derivative f'''(x)

functions Key Derivative Rules

xⁿ n·xⁿ⁻¹

sin(x) cos(x)

cos(x) −sin(x)

tan(x) sec²(x)

eˣ eˣ

ln(x) 1/x

√x 1/(2√x)

aˣ aˣ·ln(a)

rule Combination Rules

Sum / Difference

(u ± v)' = u' ± v'

Product Rule

(uv)' = u'v + uv'

Quotient Rule

(u/v)' = (u'v−uv')/v²

Chain Rule

[f(g(x))]' = f'(g)·g'

lightbulb Input Syntax

• ^ for power: x^3 = x³
• * or implicit: 3x or 3*x
• e = Euler's number ≈ 2.718
• pi = π ≈ 3.14159
• Functions: sin, cos, tan, ln, sqrt, exp...

How to Use This Calculator

Enter Your Function

Type the function you want to differentiate, e.g., x^3 + 2*x^2 - 5 or sin(x)*e^x

Select Variable

Choose the variable to differentiate with respect to (x, y, t, or n)

Choose Derivative Order

Select 1st, 2nd, or 3rd order derivative

View Result with Steps

See the derivative and the step-by-step solution showing which rules were applied

The Formula

Differentiation finds the instantaneous rate of change of a function. Key rules: Power Rule d/dx(xⁿ) = nxⁿ⁻¹, Product Rule d/dx(uv) = u'v + uv', Quotient Rule d/dx(u/v) = (u'v − uv')/v², Chain Rule d/dx[f(g(x))] = f'(g(x))·g'(x).

d/dx[f(x)] — Apply differentiation rules

lightbulb Variables Explained

f(x) The function to differentiate
f'(x) The first derivative (rate of change)
f''(x) The second derivative (rate of change of rate of change)
x The variable of differentiation

tips_and_updates Pro Tips

Use ^ for exponents: x^2 means x²

Use * for multiplication: 3*x or just 3x (implicit multiplication supported)

Supported functions: sin, cos, tan, sec, csc, cot, arcsin, arccos, arctan, exp, ln, log, sqrt, abs

Use 'e' for Euler's number (≈ 2.718) and 'pi' for π (≈ 3.14159)

Chain Rule is applied automatically: d/dx(sin(2x)) = 2cos(2x)

For the second derivative, select order 2 — it differentiates the result again

Compute Derivatives Step by Step for Any Function

Differentiation is one of the two fundamental operations in calculus, measuring how a function changes as its input changes. Derivatives appear everywhere in science and engineering — velocity is the derivative of position, acceleration is the derivative of velocity, marginal cost is the derivative of total cost, and the slope of a tangent line is the derivative of the curve at that point. Calculus students spend significant time learning differentiation rules: the power rule, product rule, quotient rule, chain rule, and the derivatives of trigonometric, exponential, and logarithmic functions. While mastering these rules is essential, verifying answers and understanding the step-by-step process is equally important for building intuition. This derivative calculator performs symbolic differentiation on any function you enter, supporting polynomials, rational functions, trigonometric functions (sin, cos, tan, sec, csc, cot), exponential and logarithmic functions, and compositions of all these via the chain rule. It shows each differentiation step with the rule applied, computes higher-order derivatives up to the fifth order, and evaluates the derivative at any specified point.

What is a Derivative Calculator?

A derivative calculator is a tool that applies calculus differentiation rules automatically to find the exact derivative of any mathematical function. Instead of manually applying the Power Rule, Chain Rule, Product Rule, or Quotient Rule, you simply enter your function and get the answer instantly with full step-by-step working.

Differentiation Rules Explained

The Power Rule (d/dx xⁿ = nxⁿ⁻¹) handles polynomials. The Chain Rule handles composite functions. The Product and Quotient Rules handle multiplication and division of functions. Our calculator applies these rules automatically and shows you exactly which rule was used at each step.

Frequently Asked Questions

Related Tools

View all Math View all arrow_forward

calculate