Integration Differentiation Calculator

Calculus is a branch of mathematics that deals with rates of change and accumulation. Integration and differentiation are two fundamental operations in calculus. This calculator helps you compute derivatives and integrals of functions, providing both the result and a graphical representation.

What is Integration and Differentiation?

Integration and differentiation are inverse operations in calculus. Differentiation finds the rate of change of a function, while integration finds the accumulation of quantities.

Differentiation

The derivative of a function measures how a function changes as its input changes. For a function f(x), the derivative f'(x) represents the slope of the tangent line at any point x.

Integration

Integration calculates the area under a curve. For a function f(x), the integral ∫f(x)dx from a to b gives the net area between the curve and the x-axis from x=a to x=b.

Note: This calculator works with basic functions. For complex functions, you may need more advanced calculus tools.

How to Use This Calculator

Select whether you want to compute a derivative or an integral.
Enter the function you want to analyze (e.g., x², sin(x), e^x).
If computing an integral, enter the lower and upper limits.
Click "Calculate" to see the result and a graphical representation.

Example Calculation

Let's find the derivative of f(x) = x³ + 2x.

f'(x) = d/dx (x³ + 2x) = 3x² + 2

The derivative is 3x² + 2.

Formulas Used

This calculator uses standard calculus formulas for differentiation and integration of basic functions.

Differentiation Formulas

Function	Derivative
f(x) = xⁿ	f'(x) = n xⁿ⁻¹
f(x) = sin(x)	f'(x) = cos(x)
f(x) = cos(x)	f'(x) = -sin(x)
f(x) = eˣ	f'(x) = eˣ

Integration Formulas

Function	Integral
f(x) = xⁿ	∫xⁿ dx = xⁿ⁺¹/(n+1) + C
f(x) = sin(x)	∫sin(x) dx = -cos(x) + C
f(x) = cos(x)	∫cos(x) dx = sin(x) + C
f(x) = eˣ	∫eˣ dx = eˣ + C

Common Applications

Integration and differentiation have numerous applications in physics, engineering, economics, and other fields.

Physics

Calculating velocity from position (differentiation)
Determining displacement from velocity (integration)
Finding work done by a variable force

Engineering

Analyzing electrical circuits
Optimizing structural designs
Modeling fluid dynamics

Economics

Calculating marginal cost and revenue
Analyzing supply and demand curves

FAQ

What is the difference between differentiation and integration?

Differentiation finds the rate of change of a function, while integration finds the accumulation of quantities. They are inverse operations in calculus.

Can this calculator handle complex functions?

This calculator works with basic functions. For complex functions, you may need more advanced calculus tools or software.

What are the common applications of integration and differentiation?

They are used in physics for motion analysis, in engineering for circuit analysis, and in economics for cost and revenue analysis.