Differentiation Calculator

Enter your function f(x):

Differentiate with respect to:

Order of derivative:

Result:

Step-by-step solution:

What is Differentiation?

Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. The derivative of a function at a particular point represents the rate of change or the slope of the tangent line to the function’s graph at that point.

Mathematically, the derivative of a function f(x) is defined as the limit:

f'(x) = lim[h→0] (f(x+h) – f(x))/h

This fundamental definition, known as the first principle of derivatives, forms the basis for all differentiation rules and techniques.

Essential Differentiation Rules

Power Rule

d/dx[x^n] = n·x^(n-1)

Example: d/dx[x³] = 3x²

Constant Rule

d/dx[c] = 0 (where c is a constant)

Example: d/dx[7] = 0

Sum and Difference Rules

d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

Example: d/dx[x² + 3x] = 2x + 3

Product Rule

d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

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

Quotient Rule

d/dx[f(x)/g(x)] = [f'(x)·g(x) – f(x)·g'(x)] / [g(x)]²

Example: d/dx[x²/(x+1)] = [2x(x+1) – x²] / (x+1)²

Chain Rule

d/dx[f(g(x))] = f'(g(x))·g'(x)

Example: d/dx[sin(x²)] = cos(x²)·2x

Common Function Derivatives

Trigonometric Functions

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

Exponential and Logarithmic Functions

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

Inverse Trigonometric Functions

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

Applications of Derivatives

Physics and Engineering

In physics, derivatives represent rates of change. Velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity. In engineering, derivatives help analyse system behaviour and optimise designs.

Economics and Finance

Marginal cost and marginal revenue are derivatives that help businesses optimise production and pricing strategies. In finance, derivatives of price functions help calculate rates of return and risk measures.

Biology and Medicine

Population growth rates, drug concentration changes, and enzyme reaction rates are all modelled using derivatives. These applications help scientists make predictions and optimise treatments.

How to Use This Calculator

Enter your function: Type your mathematical function in standard notation. Use * for multiplication, ^ for powers, and standard function names like sin, cos, ln, exp.
Select the variable: Choose which variable you want to differentiate with respect to (usually x).
Choose the order: Select whether you want the first, second, or third derivative.
Click Calculate: The calculator will compute the derivative and show detailed steps.
Review the solution: Examine both the final answer and the step-by-step working to understand the process.

Frequently Asked Questions

What functions can this calculator differentiate?

This calculator can handle polynomial functions, trigonometric functions (sin, cos, tan), exponential functions (e^x, a^x), logarithmic functions (ln, log), inverse trigonometric functions, and combinations of these using arithmetic operations.

How accurate are the results?

The calculator uses proven mathematical algorithms to ensure 100% accuracy for all supported functions. Results are computed using symbolic mathematics, not numerical approximations.

Can I calculate higher-order derivatives?

Yes, this calculator supports first, second, and third derivatives. Higher-order derivatives are computed by repeatedly applying differentiation rules.

What if my function contains multiple variables?

The calculator treats functions with multiple variables as partial derivatives, differentiating with respect to your selected variable while treating others as constants.

How should I format my input?

Use standard mathematical notation: x^2 for x squared, sin(x) for sine of x, ln(x) for natural logarithm, e^x for exponential. Always use * for multiplication between variables and constants.