differentiation calculator

What this differentiation calculator does

This tool computes symbolic derivatives for functions of one variable, x. It can return the first derivative, second derivative, or higher-order derivatives, and it can optionally evaluate the result at a specific point. If you are studying calculus, checking homework, or validating a model, this gives you a fast and reliable way to verify your derivative expressions.

How to use it

Step-by-step

• Enter your function in the f(x) box.
• Select the derivative order (1 through 8).
• Optionally enter a value for x to evaluate the derivative numerically.
• Click Differentiate to see results instantly.

Input tips

• Use * for multiplication: 2*x (not 2x).
• Use ^ for powers: x^5.
• Natural log is ln(x).
• Constants like pi and e are supported.

Why differentiation matters

Derivatives measure how quickly one quantity changes with respect to another. That idea appears in almost every technical field:

• Physics: velocity and acceleration come from derivatives of position.
• Economics: marginal cost and marginal revenue are derivatives.
• Machine learning: gradient-based optimization relies on derivatives.
• Engineering: sensitivity and rate-of-change analysis are core design tools.

Common derivative rules (quick reference)

• Power Rule: d/dx [x^n] = n*x^(n-1)
• Constant Multiple: d/dx [c*f(x)] = c*f'(x)
• Sum 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'g - fg')/g^2
• Chain Rule: d/dx [f(g(x))] = f'(g(x))*g'(x)

Worked examples

Example 1: Polynomial

For f(x) = x^3 + 2*x^2 - 5*x + 1, the first derivative is f'(x) = 3*x^2 + 4*x - 5. At x = 2, the slope is 15.

Example 2: Trigonometric function

For f(x) = sin(x), we get f'(x) = cos(x). This tells us the slope of sine at any point is exactly cosine at that same point.

Example 3: Exponential + logarithmic

For f(x) = e^(2*x) + ln(x), the derivative is f'(x) = 2*e^(2*x) + 1/x. This combines the chain rule and log derivative rule.

Common mistakes to avoid

• Forgetting parentheses in composite expressions (for example, write sin(2*x)).
• Using implicit multiplication where explicit multiplication is required.
• Entering invalid domains (such as ln(x) with a negative x value).
• Confusing function value with derivative value at a point.

Final note

A calculator is a powerful assistant, but your understanding is still the main skill. Use this tool to check answers, explore patterns, and build intuition about slopes, curvature, and change.