Derivative Calculator
Instantly compute the derivative of almost any mathematical expression. Whether you're working on a calculus homework assignment or studying advanced mathematics, our Derivative Calculator provides accurate real-time results for polynomials, trigonometric functions, exponentials, and more.
Need to find a derivative quickly? Enter your function, specify the variable, and get the instant symbolic derivative. Perfect for checking your calculus work.
- Instant real-time evaluation
- Supports multiple derivative orders (1st, 2nd, 3rd)
- 100% free, no software installation needed
Introduction to Derivatives
In calculus, the derivative measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). It is a fundamental tool of calculus used to find rates of change and slopes of curves.
Our online Derivative Calculator automates this process by applying standard differentiation rules—such as the power rule, product rule, quotient rule, and chain rule—to provide you with an exact analytical expression of the derivative.
How to Use the Derivative Calculator
Using this tool is straightforward and designed for instant feedback. Follow these steps:
- Enter the Function: Type your mathematical expression in the "Function f(x)" field. You can use standard notation like `x^2`, `sin(x)`, `exp(x)`, or `sqrt(x)`.
- Specify the Variable: Ensure the "Variable" field matches the variable you want to differentiate with respect to (usually `x`).
- Select the Order: Choose whether you want the first derivative (f'), second derivative (f''), or third derivative (f''').
- Read the Result: The derivative will be calculated automatically and displayed in the result box.
Common Differentiation Rules
The calculator internally applies several foundational rules of calculus:
- Power Rule: For `f(x) = x^n`, the derivative is `n * x^(n-1)`.
- Sum Rule: The derivative of `f(x) + g(x)` is `f'(x) + g'(x)`.
- Product Rule: The derivative of `f(x) * g(x)` is `f'(x)g(x) + f(x)g'(x)`.
- Chain Rule: The derivative of `f(g(x))` is `f'(g(x)) * g'(x)`.
Practical Examples
1. Polynomials
Function: 3x^3 - 2x^2 + 5x
Derivative: 9x^2 - 4x + 5
2. Trigonometry
Function: sin(x) + cos(x)
Derivative: cos(x) - sin(x)
3. Exponentials
Function: e^(2x)
Derivative: 2 * e^(2x)
Frequently Asked Questions
What is a second derivative?
A second derivative is simply the derivative of the first derivative. It measures how the rate of change itself is changing (e.g., acceleration is the second derivative of position).
Does this tool support partial derivatives?
Currently, the tool evaluates ordinary derivatives with respect to the single variable you specify in the input fields.
How do I write square roots?
You can type `sqrt(x)` or `x^(1/2)` to represent a square root in the function input box.
Conclusion
Mastering calculus requires understanding how functions change. Our Derivative Calculator acts as an instant reference tool to check your work or quickly solve complex mathematical problems. Keep it handy whenever you're dealing with rates of change or optimization problems in math and physics.