Derivative Calculator

Calculate derivatives of functions with detailed step-by-step solutions and explanations

Function Input

Enter function f(x):

Use * for multiplication, ^ for powers, and standard functions like sin, cos, ln, exp

Differentiate with respect to:

Derivative order:

Quick function templates:

Display options:

Show step-by-step solution Simplify result Show derivative rules used

Function Syntax:

Power: x^2, x^3

Multiplication: 2*x, 3*x^2

Division: x/2, 1/x

Trig: sin(x), cos(x), tan(x)

Log: ln(x), log(x)

Exp: exp(x), e^x

Results

Enter a function and click "Calculate Derivative" to see the result

Understanding Derivatives

What is a Derivative?

A derivative represents the rate of change of a function with respect to a variable. Geometrically, it's the slope of the tangent line to the curve at any point.

Mathematical Definition

f'(x) = lim[h→0] [f(x+h) - f(x)] / h

The limit of the difference quotient as h approaches zero

Derivative Notation

There are several common notations for derivatives, each with its own advantages in different contexts.

Common Notations

Leibniz: df/dx, d²f/dx²

Lagrange: f'(x), f''(x)

Newton: ḟ, f̈ (for time)

Operator: Df, D²f

Basic Derivative Rules

Power Rule

d/dx [xⁿ] = n·x^(n-1)

Most fundamental rule for polynomials

Product Rule

d/dx [f·g] = f'·g + f·g'

For derivatives of products

Quotient Rule

d/dx [f/g] = (f'·g - f·g')/g²

For derivatives of quotients

Chain Rule

d/dx [f(g(x))] = f'(g(x))·g'(x)

For composite functions

Constant Rule

d/dx [c] = 0

Derivative of constants is zero

Sum Rule

d/dx [f + g] = f' + g'

Derivative of sum is sum of derivatives

Common Function Derivatives

Trigonometric

d/dx [sin x] = cos x

d/dx [cos x] = -sin x

d/dx [tan x] = sec²x

Exponential

d/dx [eˣ] = eˣ

d/dx [aˣ] = aˣ ln a

d/dx [xˣ] = xˣ(ln x + 1)

Logarithmic

d/dx [ln x] = 1/x

d/dx [log x] = 1/(x ln 10)

d/dx [logₐ x] = 1/(x ln a)

Applications

• Physics: velocity, acceleration
• Economics: marginal analysis
• Optimization problems
• Rate of change analysis
• Curve sketching
• Related rates

Higher-Order Derivatives

• Second derivative: concavity
• Third derivative: jerk in physics
• nth derivative: Taylor series
• Critical points analysis
• Inflection points

Common Mistakes

• Forgetting chain rule
• Incorrect product rule application
• Sign errors in quotient rule
• Not simplifying results
• Confusing d/dx notation

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