What Is a Derivative?
The derivative of a function measures how fast its output changes as its input changes — its instantaneous rate of change, or the slope of the tangent line at any point on its graph. If f(x) describes a position, f'(x) describes velocity; if f(x) describes velocity, f'(x) describes acceleration.
How to Use This Derivative Calculator
- Type your function of x into the f(x) field, using * for multiplication and ^ for powers (e.g.
3*x^2 + sin(x)). - Choose whether you want the 1st, 2nd, or 3rd derivative.
- Optionally enter a value of x to evaluate the derivative at that exact point.
- Click Differentiate to see the simplified result and every intermediate step.
Differentiation Rules This Calculator Uses
Behind the scenes, the calculator builds a mathematical expression tree from your function and applies the standard rules of calculus automatically:
Worked Examples
Bring the exponent down and reduce it by one: f'(x) = 2x
The derivative of sine is cosine: f'(x) = cos(x)
The natural exponential is its own derivative: f'(x) = eˣ
Understanding Higher-Order Derivatives
The second derivative f''(x) is the derivative of f'(x) — it measures how the rate of change itself is changing (for a position function, this is acceleration). The third derivative f'''(x) goes one step further. This calculator applies the differentiation rules repeatedly and re-simplifies after every step, so higher-order results stay as clean as possible.
Common Mistakes to Avoid
- Forgetting the chain rule when differentiating a function inside another function, like sin(2x) or (x²+1)³.
- Applying the power rule to a sum term-by-term but forgetting a constant multiplier, e.g. treating 5x² like x².
- Mixing up the product rule and quotient rule signs — the quotient rule subtracts, the product rule adds.
Frequently Asked Questions
A derivative measures the instantaneous rate of change of a function — geometrically, the slope of the line tangent to its graph at a given point. It's the foundation of differential calculus.
The power rule says that for f(x) = xⁿ, the derivative is f'(x) = n·x⁽ⁿ⁻¹⁾ — bring the exponent down as a multiplier and reduce the exponent by one. It's the most common rule used when differentiating polynomials.
The chain rule differentiates a "function of a function," like sin(3x) or (x²+1)⁵. It says d/dx[f(g(x))] = f'(g(x))·g'(x) — differentiate the outer function, keep the inner function the same, then multiply by the derivative of the inner function.
Not currently. This tool differentiates explicit single-variable functions of the form y = f(x). Implicit differentiation (for equations like x² + y² = 25) and multivariable calculus (partial derivatives) are outside its current scope.
They're inverse operations. A derivative measures a function's rate of change, while an antiderivative (or indefinite integral) reverses that process to recover a function from its rate of change. Try our Integral Calculator to go the other direction.