Derivative Calculator
- Write the function: f(x) = x³ + 2x² + 5.
- Rules needed: Sum rule, constant rule, constant multiple rule, power rule.
- Differentiate with respect to x: f′(x) = 3x² + 4x.
- Substitute x = 2: f′(2) = 20.
- Sum rule: differentiate each term on its own, then add the results.
- Constant rule: a term with no variable has a derivative of 0.
- Constant multiple rule: a number multiplying a function stays in front.
- Power rule: the derivative of xⁿ is n·xⁿ⁻¹.
- Numerical check at x = 2: the slope of f over a tiny step is about 20, which matches 20.
- The tangent line touches the curve at (2, 21) with slope 20.
Derivative Calculator: Differentiate Any Function with Steps
This free derivative calculator works out the derivative of a function you type in and shows how it got there. It handles polynomials, fractions, roots, powers, trigonometric functions, exponentials and logarithms, and it can differentiate up to four times. Beside the answer you get the value of the function and its slope at a point of your choice, the equation of the tangent line and a graph that updates as you type.
Quick answer: the derivative of x³ + 2x² + 5 is 3x² + 4x. At x = 2 the slope is 20, the function value is 21 and the tangent line is y = 20x − 19.
What Is a Derivative?
A derivative tells you how quickly something changes. If a function describes the position of a car, its derivative is the speed. If it describes the cost of making goods, its derivative is the extra cost of one more unit. On a graph, the derivative at a point is the slope of the curve there, found by zooming in until the curve looks like a straight line.
Basic Differentiation Rules
Most derivatives come from a few rules that are combined. The power rule turns xⁿ into n·xⁿ⁻¹. The sum rule lets you differentiate a long expression one term at a time, and a number multiplying a function simply stays in front. The product rule handles two functions multiplied together, the quotient rule handles one divided by another, and the chain rule handles a function inside another function, such as sin(3x) or (x² + 1)³. The table shows a few examples of each.
| Function | Derivative | Rule used |
|---|---|---|
| x⁴ | 4x³ | Power rule |
| 5x² + 3x | 10x + 3 | Sum and constant multiple |
| sin(x) | cos(x) | Trigonometric rule |
| e^(2x) | 2e^(2x) | Exponential and chain rule |
| ln(x) | 1/x | Logarithm rule |
| x² sin(x) | x² cos(x) + 2x sin(x) | Product rule |
| x/(x + 1) | 1/(x + 1)² | Quotient rule |
| (x² + 1)³ | 6x(x² + 1)² | Chain rule |
How to Enter a Function
Type the function the way you would write it on paper. Use ^ for powers, so x squared is x^2, and you can leave out the multiplication sign, so 3x and 2sin(x) both work. Write square roots as sqrt(x), the exponential function as e^x and pi for π. Functions such as sin, cos, tan, ln and log need brackets around their input. The buttons below the box insert common symbols without typing. Letters other than the chosen variable are treated as constants, so differentiating a·x² with respect to x gives 2a·x.
Higher-Order Derivatives
Differentiating the derivative gives the second derivative, which describes how the slope itself changes. A positive second derivative means the curve bends upward, a negative one means it bends downward, and a change of sign marks an inflection point. In physics, the first derivative of position is velocity and the second is acceleration. Pick the 2nd, 3rd or 4th order to repeat the process automatically.
Slope, Tangent Line and the Graph
Enter a point to see the slope of the curve there. The green dashed line on the graph is the tangent line, which touches the curve at that point and has the same slope. Where the slope is zero, the curve is flat, which often marks a highest or lowest point. For finding where a quadratic reaches its peak, the quadratic calculator is a useful partner, and the polynomial calculator helps with factoring and roots of higher-degree expressions. To go the other way, from a rate of change back to the original function, use the integral calculator.
Who Uses a Derivative Calculator?
Students in calculus and pre-calculus use it to check homework and to see each stage of the working. Engineers, economists and scientists use derivatives to find rates of change and to optimize a quantity. For arithmetic that comes up along the way, the scientific calculator and the math solver can help.
How to Use This Derivative Calculator
Type your function
Enter it in the box, or tap the shortcut buttons to insert powers, roots and functions.
Choose the variable and order
Pick x, y, z or t, then 1st to 4th derivative.
Set a point
Type a value or drag the slider to see the slope and tangent line there.
Read the results
See the derivative, the rules used, the step-by-step working and the graph.