Derivative Calculator

Calculate derivatives for any function with detailed steps. Free online Derivative Calculator with solving steps for calculus students.

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d/dx

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What is a Derivative?

A derivative represents the instantaneous rate of change of a function with respect to a variable. It's one of the fundamental concepts in calculus, measuring how a function's output changes as its input changes.

Geometric Interpretation

Geometrically, the derivative at a point represents the slope of the tangent line to the function's graph at that point. A larger derivative means a steeper slope; a derivative of zero means no change (horizontal tangent).

Notation

Derivatives can be written in several forms:

  • Prime notation: f'(x)
  • Leibniz notation: df/dx or dy/dx
  • Operator notation: Df(x) or D_x f

All represent the same concept: the rate at which f changes with respect to x.

Common Derivative Rules

Power Rule

d/dx(x^n) = n·x^(n-1)

The power rule is the most fundamental derivative rule for polynomials.

Examples:

  • d/dx(x²) = 2x
  • d/dx(x³) = 3x²
  • d/dx(x^5) = 5x^4

Constant Rule

d/dx(c) = 0

The derivative of any constant is zero.

Examples:

  • d/dx(5) = 0
  • d/dx(π) = 0

Constant Multiple Rule

d/dx(c·f(x)) = c·f'(x)

Constants can be pulled out of derivatives.

Example: d/dx(3x²) = 3·d/dx(x²) = 3·2x = 6x

Sum and Difference Rules

d/dx(f ± g) = f' ± g'

Derivatives of sums/differences are sums/differences of derivatives.

Example: d/dx(x² + 3x) = d/dx(x²) + d/dx(3x) = 2x + 3

Product Rule

d/dx(f·g) = f'·g + f·g'

For products of two functions.

Example: d/dx(x²·sin(x)) = 2x·sin(x) + x²·cos(x)

Quotient Rule

d/dx(f/g) = (f'·g - f·g') / g²

For quotients of two functions.

Example: d/dx(x/(x+1)) = (1·(x+1) - x·1) / (x+1)² = 1/(x+1)²

Chain Rule

d/dx(f(g(x))) = f'(g(x))·g'(x)

For composite functions (function within a function).

Example: d/dx(sin(x²)) = cos(x²)·2x = 2x·cos(x²)

Trigonometric Functions

  • d/dx(sin x) = cos x
  • d/dx(cos x) = -sin x
  • d/dx(tan x) = sec²x
  • d/dx(cot x) = -csc²x
  • d/dx(sec x) = sec x·tan x
  • d/dx(csc x) = -csc x·cot x

Exponential Functions

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

Logarithmic Functions

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

Step-by-Step Examples

Example 1: Polynomial (Power Rule)

Problem: Find the derivative of f(x) = x³ + 2x² - 5x + 7

Solution:

  1. Apply power rule to each term:
    • d/dx(x³) = 3x²
    • d/dx(2x²) = 4x
    • d/dx(-5x) = -5
    • d/dx(7) = 0
  2. Combine results:
    • f'(x) = 3x² + 4x - 5

Answer: f'(x) = 3x² + 4x - 5

Example 2: Product Rule

Problem: Find the derivative of f(x) = x²·sin(x)

Solution:

  1. Identify: f = x², g = sin(x)
  2. Find derivatives: f' = 2x, g' = cos(x)
  3. Apply product rule: f'·g + f·g'
  4. Calculate:
    • = 2x·sin(x) + x²·cos(x)

Answer: f'(x) = 2x·sin(x) + x²·cos(x)

Example 3: Chain Rule

Problem: Find the derivative of f(x) = sin(x²)

Solution:

  1. Identify outer function: sin(u) where u = x²
  2. Outer derivative: cos(u)
  3. Inner derivative: d/dx(x²) = 2x
  4. Apply chain rule:
    • f'(x) = cos(x²)·2x = 2x·cos(x²)

Answer: f'(x) = 2x·cos(x²)

Example 4: Quotient Rule

Problem: Find the derivative of f(x) = (x+1)/(x-1)

Solution:

  1. Identify: numerator (x+1), denominator (x-1)
  2. Derivatives: d/dx(x+1) = 1, d/dx(x-1) = 1
  3. Apply quotient rule:
    • = (1·(x-1) - (x+1)·1) / (x-1)²
    • = (x - 1 - x - 1) / (x-1)²
    • = -2 / (x-1)²

Answer: f'(x) = -2/(x-1)²

Applications of Derivatives

1. Finding Maximum and Minimum Values

Derivatives help identify critical points where f'(x) = 0 or is undefined. These points are candidates for local maxima and minima.

Example: To find the maximum profit, set the derivative of the profit function to zero and solve.

2. Velocity and Acceleration in Physics

  • Velocity is the derivative of position: v(t) = s'(t)
  • Acceleration is the derivative of velocity: a(t) = v'(t) = s''(t)

Example: If position s(t) = -4.9t² + 20t, then velocity v(t) = -9.8t + 20.

3. Optimization Problems

Derivatives are crucial in finding optimal solutions in engineering, economics, and business.

Example: Minimizing material costs, maximizing production efficiency, finding optimal pricing.

4. Rate of Change Analysis

Derivatives quantify how quickly quantities change in real-world scenarios.

Examples:

  • Population growth rates
  • Chemical reaction rates
  • Temperature change over time

Frequently Asked Questions

What does d/dx mean?

The notation d/dx is an operator that means "take the derivative with respect to x." It tells us we're finding the rate of change of a function as x varies.

When do I use which derivative rule?

  • Single variable term (x², x³): Power rule
  • Product of functions (x·sin(x)): Product rule
  • Quotient of functions (x/(x+1)): Quotient rule
  • Composition of functions (sin(x²)): Chain rule
  • Sum/difference (x² + 3x): Apply rules term-by-term

How do I find higher-order derivatives?

Simply differentiate multiple times:

  • First derivative: f'(x)
  • Second derivative: f''(x) = d/dx(f'(x))
  • Third derivative: f'''(x) = d/dx(f''(x))

Example: If f(x) = x³, then f'(x) = 3x², f''(x) = 6x, f'''(x) = 6.

What's the difference between dy/dx and ∂y/∂x?

  • dy/dx: Ordinary derivative for functions of one variable
  • ∂y/∂x: Partial derivative for functions of multiple variables (holds other variables constant)

Why is the derivative of e^x equal to itself?

The natural exponential function e^x has the unique property that its rate of change equals its value at every point. This makes it incredibly important in mathematics, especially for modeling growth and decay.

How do I remember trigonometric derivatives?

Memory trick:

  • sin → cos (positive)
  • cos → -sin (negative)
  • tan → sec² (squared secant)

Remember: derivatives of "co-functions" (cos, cot, csc) have negative signs.

References

This calculator uses standard calculus principles verified against authoritative mathematics resources: