Derivative Calculator

Function f(x)

Supported: polynomials (x^2), trig functions (sin, cos, tan), e^x, ln(x), sqrt(x)

Variable

Quick Examples

Derivative Calculator: Complete Calculus Guide

A derivative represents the instantaneous rate of change of a function with respect to its variable.Geometrically, the derivative at a point gives the slope of the tangent line to the function's curve at that point. Derivatives are fundamental to calculus and essential for optimization, physics, engineering, and mathematical modeling.

Our professional derivative calculator provides step-by-step differentiation using standard calculus rules including the power rule, product rule, quotient rule, and chain rule. The calculator supports polynomials, trigonometric functions, exponential functions, logarithms, and combinations thereof with interactive visualization.

Quick Answer

To find a derivative: Enter your function f(x), select the variable for differentiation, and click calculate. The result shows f'(x) with the differentiation rule used. For example, the derivative of x² + 3x + 1 is 2x + 3 using the power rule and sum rule.

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Mathematical Foundation

f'(x) = lim_h→0 [f(x+h) - f(x)] / h

The derivative definition using the limit of the difference quotient

Key Differentiation Concepts:

Geometric Interpretation

The derivative f'(a) represents the slope of the tangent line to the curve y = f(x) at the point (a, f(a)). Positive derivatives indicate increasing functions, negative derivatives indicate decreasing functions.

Rate of Change

Derivatives measure instantaneous rates of change. In physics, the derivative of position with respect to time gives velocity; the derivative of velocity gives acceleration.

Critical Points

Points where f'(x) = 0 or f'(x) is undefined are critical points. These often correspond to local maxima, minima, or inflection points of the original function.

Essential Differentiation Rules

Basic Rules

Fundamental rules for differentiating simple functions.

Constant Rule: d/dx(c) = 0
Power Rule: d/dx(x^n) = n·x^(n-1)
Sum Rule: d/dx(f + g) = f' + g'

Best for: Polynomials, simple algebraic functions

Examples: d/dx(x³) = 3x², d/dx(5x² + 3x - 7) = 10x + 3

Product and Quotient Rules

Rules for differentiating products and quotients of functions.

Product Rule: d/dx(f·g) = f'·g + f·g'
Quotient Rule: d/dx(f/g) = (f'·g - f·g')/g²

Best for: Products and ratios of functions

Examples: d/dx(x²·sin(x)), d/dx(ln(x)/x²)

Chain Rule

Rule for differentiating composite functions.

Chain Rule: d/dx(f(g(x))) = f'(g(x))·g'(x)

Best for: Composite functions, nested operations

Examples: d/dx(sin(x²)) = cos(x²)·2x, d/dx(e^(3x)) = e^(3x)·3

Derivatives of Common Functions

Trigonometric Functions

Basic Trig Derivatives

d/dx(sin(x)) = cos(x)
d/dx(cos(x)) = -sin(x)
d/dx(tan(x)) = sec²(x)

Inverse Trig Functions

d/dx(arcsin(x)) = 1/√(1-x²)
d/dx(arccos(x)) = -1/√(1-x²)
d/dx(arctan(x)) = 1/(1+x²)

Exponential & Logarithmic

Exponential Functions

d/dx(e^x) = e^x
d/dx(a^x) = a^x · ln(a)
d/dx(x^x) = x^x(ln(x) + 1)

Logarithmic Functions

d/dx(ln(x)) = 1/x
d/dx(log_a(x)) = 1/(x·ln(a))
d/dx(log(x)) = 1/(x·ln(10))

Example Problems with Solutions

Example 1: Polynomial Function

Find the derivative of f(x) = 3x⁴ - 2x³ + 5x - 7

f(x) = 3x⁴ - 2x³ + 5x - 7

Apply power rule to each term:

d/dx(3x⁴) = 3 · 4x³ = 12x³

d/dx(-2x³) = -2 · 3x² = -6x²

d/dx(5x) = 5 · 1 = 5

d/dx(-7) = 0

f'(x) = 12x³ - 6x² + 5

Answer: f'(x) = 12x³ - 6x² + 5

Example 2: Trigonometric Function

Find the derivative of g(x) = sin(x) · cos(x)

g(x) = sin(x) · cos(x)

Apply product rule: (f·g)' = f'·g + f·g'

f = sin(x), f' = cos(x)

g = cos(x), g' = -sin(x)

g'(x) = cos(x)·cos(x) + sin(x)·(-sin(x))

g'(x) = cos²(x) - sin²(x)

g'(x) = cos(2x)

Answer: g'(x) = cos(2x)

Example 3: Chain Rule Application

Find the derivative of h(x) = ln(x² + 1)

h(x) = ln(x² + 1)

Apply chain rule: (f(g(x)))' = f'(g(x))·g'(x)

Outer function: f(u) = ln(u), f'(u) = 1/u

Inner function: g(x) = x² + 1, g'(x) = 2x

h'(x) = 1/(x² + 1) · 2x

h'(x) = 2x/(x² + 1)

Answer: h'(x) = 2x/(x² + 1)

Function Input Syntax Guide

Basic Operations

Addition:x + 5

Subtraction:x - 3

Multiplication:3*x or 3x

Division:x / 2

Power:x^2, x^3

Mathematical Functions

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

Exponential:exp(x), e^x

Logarithm:ln(x), log(x)

Square Root:sqrt(x)

Absolute:abs(x)

Important Notes

• Use parentheses to ensure correct order of operations
• Variable must be lowercase 'x', 'y', 't', or 'u'
• Constants: pi (π ≈ 3.14159), e (≈ 2.71828)
• Multiplication can be implicit: 3x means 3*x
• Complex expressions may require specific differentiation rules

Real-World Applications of Derivatives

Physics & Engineering

Motion Analysis

Velocity: First derivative of position v(t) = s'(t)
Acceleration: Second derivative a(t) = s''(t)

Optimization

Find maximum efficiency, minimize material usage, optimal design parameters

Rate Problems

Heat transfer rates, population growth rates, chemical reaction rates

Economics & Business

Marginal Analysis

Marginal cost, marginal revenue, marginal profit calculations

Profit Maximization

Find optimal production levels, pricing strategies, resource allocation

Elasticity

Price elasticity of demand, income elasticity, supply elasticity

Frequently Asked Questions

What is a derivative?

A derivative measures the instantaneous rate of change of a function. Geometrically, it represents the slope of the tangent line to the function's curve at any given point. Derivatives tell us how quickly a function is changing and in what direction.

What are the basic differentiation rules?

The main rules are: Power Rule (d/dx(x^n) = nx^(n-1)), Product Rule((fg)' = f'g + fg'), Quotient Rule ((f/g)' = (f'g - fg')/g²), and Chain Rule((f(g(x)))' = f'(g(x))·g'(x)). These cover most differentiation scenarios.

How do I find critical points?

Critical points occur where f'(x) = 0 or f'(x) is undefined. To find them: (1) Calculate the derivative, (2) Set f'(x) = 0 and solve for x, (3) Find where f'(x) is undefined. These points are candidates for local maxima, minima, or inflection points.

What's the difference between f'(x) and df/dx?

Both notations represent the same concept - the derivative of f with respect to x. f'(x)is Lagrange notation, while df/dx is Leibniz notation. They're interchangeable and represent the instantaneous rate of change.

When do I use the chain rule?

Use the chain rule for composite functions - functions inside other functions. Examples include sin(x²), e^(3x), ln(x² + 1), or (x² + 1)^5. The pattern is: if you have f(g(x)), then the derivative is f'(g(x)) · g'(x).

What are higher-order derivatives?

Higher-order derivatives are derivatives of derivatives. The second derivative f''(x) gives concavity information and acceleration in physics. The third derivative f'''(x) relates to the rate of change of acceleration (jerk). Each successive derivative provides deeper insights into function behavior.