Derivative & Integral Calculator

What is a Derivative & Integral Calculator?

A derivative and integral calculator is a symbolic mathematics tool that computes derivatives (rates of change) and integrals (area under curves) of mathematical functions. Unlike numerical calculators, it provides exact symbolic solutions and can show step-by-step working.

This calculator handles a wide range of functions including polynomials, trigonometric functions, exponential and logarithmic functions, and their combinations. It's an essential tool for students learning calculus and professionals working with mathematical models.

Derivatives

Definition

The derivative of a function f(x) represents the instantaneous rate of change of the function with respect to its variable. It is defined as:

f'(x) = lim[h→0] (f(x+h) - f(x)) / h

Common Derivative Rules

• Power Rule: d/dx(xⁿ) = n·xⁿ⁻¹
• Product Rule: d/dx(f·g) = f'·g + f·g'
• Quotient Rule: d/dx(f/g) = (f'·g - f·g') / g²
• Chain Rule: d/dx(f(g(x))) = f'(g(x))·g'(x)

Trigonometric Derivatives

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

Exponential & Logarithmic Derivatives

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

Integrals

Definition

The integral of a function f(x) represents the area under the curve from a starting point to an ending point. The indefinite integral (antiderivative) is a function whose derivative is f(x):

∫f(x)dx = F(x) + C, where F'(x) = f(x)

Common Integral Rules

• Power Rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
• Sum Rule: ∫(f + g)dx = ∫f dx + ∫g dx
• Constant Multiple: ∫k·f dx = k·∫f dx

Trigonometric Integrals

• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C
• ∫sec² x dx = tan x + C

Exponential & Logarithmic Integrals

• ∫eˣ dx = eˣ + C
• ∫(1/x) dx = ln|x| + C
• ∫aˣ dx = aˣ/ln(a) + C

Applications of Derivatives and Integrals

Derivatives and integrals have countless real-world applications:

• Physics: Velocity and acceleration (derivatives), displacement (integrals)
• Economics: Marginal cost and revenue (derivatives), total cost (integrals)
• Engineering: Optimization problems, stress analysis, signal processing
• Biology: Population growth rates, drug concentration modeling
• Computer Graphics: Curve interpolation and animation
• Machine Learning: Gradient descent optimization
• Statistics: Probability density functions and cumulative distributions

Function Syntax Guide

Use the following syntax to enter functions:

• Basic operations: +, -, *, /, ^ (power)
• Functions: sin(x), cos(x), tan(x), exp(x), ln(x), log(x), sqrt(x), abs(x)
• Constants: e (Euler's number), pi
• Use parentheses to group operations: (x+1)^2
• Use explicit multiplication: write 2*x, not 2x

Tips for Using the Calculator

• Always use explicit multiplication symbols (write 2*x, not 2x)
• Use parentheses to make the order of operations clear
• For trigonometric functions, the argument is in radians
• Check your result by differentiating an integral (should get the original function)
• Remember that indefinite integrals include an arbitrary constant C