Derivative Calculator Integral

This derivative and integral calculator helps you find the rate of change of a function (derivative) and the accumulation of quantities (integral). Whether you're studying calculus or need to solve real-world problems, this tool provides accurate results with step-by-step explanations.

What is a Derivative?

The derivative of a function measures how a quantity changes as another quantity changes. In simpler terms, it's the slope of the tangent line to the function's curve at a given point. Derivatives are essential in physics, engineering, and economics for analyzing rates of change.

Derivative Formula:

If y = f(x), then the derivative dy/dx is calculated as the limit of Δy/Δx as Δx approaches 0.

Common Derivative Rules

Power Rule: d/dx (xⁿ) = n xⁿ⁻¹
Sum/Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
Product Rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Quotient Rule: d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]²

What is an Integral?

An integral represents the area under a curve between two points. It can be used to find the accumulation of quantities, such as area under a velocity-time graph to determine distance traveled. Integrals are fundamental in calculus and have applications in physics, engineering, and economics.

Integral Formula:

∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration.

Common Integral Techniques

Substitution Method: Used for integrals of composite functions.
Integration by Parts: Used for products of functions, based on the product rule for differentiation.
Partial Fractions: Used for rational functions.

How to Use This Calculator

Select whether you want to calculate a derivative or integral.
Enter the function you want to differentiate or integrate.
If calculating a derivative, specify the point at which to find the slope.
Click "Calculate" to see the result and a graphical representation.
Review the step-by-step solution and formula used.

This calculator supports basic functions and simple derivatives/integrals. For complex functions, you may need to use more advanced mathematical software.

Formula Explanation

The calculator uses standard calculus formulas to compute derivatives and integrals. For derivatives, it applies the power rule, product rule, and other differentiation techniques. For integrals, it uses substitution, integration by parts, and other integration methods.

Example Derivative Calculation:

If f(x) = 3x² + 2x + 1, then f'(x) = 6x + 2.

Example Integral Calculation:

∫ (3x² + 2x + 1) dx = x³ + x² + x + C.

Example Calculation

Let's find the derivative of f(x) = 2x³ - 3x² + 5x - 7 at x = 2.

Apply the power rule to each term:
- d/dx (2x³) = 6x²
- d/dx (-3x²) = -6x
- d/dx (5x) = 5
- d/dx (-7) = 0
Combine the results: f'(x) = 6x² - 6x + 5.
Evaluate at x = 2: f'(2) = 6(4) - 6(2) + 5 = 24 - 12 + 5 = 17.

The derivative at x = 2 is 17.

FAQ

What is the difference between a derivative and an integral?: A derivative measures the rate of change of a function, while an integral calculates the accumulation of quantities.
Can this calculator handle complex functions?: This calculator supports basic functions and simple derivatives/integrals. For complex functions, consider using more advanced mathematical software.
What are the common rules for differentiation?: The common rules include the power rule, sum/difference rule, product rule, and quotient rule.
What are the common techniques for integration?: The common techniques include substitution, integration by parts, and partial fractions.
How can I verify the results from this calculator?: You can verify the results by manually applying the differentiation or integration rules and comparing them with the calculator's output.