Derivative Calculator with Steps
Use this free derivative calculator with steps to find the first derivative, second derivative, or third derivative of common calculus functions. Enter a function such as x^2 + 3x, sin(x), e^x, ln(x), or 1/x and get a clear answer with a step-by-step explanation.
Calculus Derivative Calculator
Enter a function below to find the derivative. This derivative solver supports common student-level expressions, including polynomials, basic trigonometric functions, exponential functions, natural logarithms, square roots, and reciprocal functions.
Derivative Result
Step-by-Step Solution
How to Use the Derivative Calculator
This derivative calculator with steps is designed for US students, AP Calculus students, college learners, teachers, and tutors who need a quick way to check common calculus problems. To use the tool, enter your function, choose the variable, select the first derivative, second derivative, or third derivative, optionally enter a value, and click the calculate button.
The calculator will show the original function, the derivative answer, the derivative order, the rule used, and a simple step-by-step solution. For supported expressions, it can also evaluate the derivative at a selected point.
What Is a Derivative?
A derivative measures how a function changes. In calculus, the derivative represents the slope of a curve at a specific point. It is used to understand rate of change, motion, optimization, economics, physics, engineering, and many real-world problems.
For example, if a function describes the position of an object over time, the first derivative can describe velocity. The second derivative can describe acceleration. This is why derivatives are important in AP Calculus, college calculus, physics, business, economics, and engineering.
Common Derivative Rules
| Rule Name | Formula | Example |
|---|---|---|
| Constant Rule | d/dx(c) = 0 | d/dx(8) = 0 |
| Power Rule | d/dx(x^n) = nx^(n-1) | d/dx(x^3) = 3x^2 |
| Constant Multiple Rule | d/dx(cf(x)) = c f'(x) | d/dx(4x^2) = 8x |
| Sum Rule | d/dx(f(x) + g(x)) = f'(x) + g'(x) | d/dx(x^2 + x) = 2x + 1 |
| Difference Rule | d/dx(f(x) - g(x)) = f'(x) - g'(x) | d/dx(x^3 - x) = 3x^2 - 1 |
| Product Rule | d/dx(fg) = f'g + fg' | d/dx(x sin x) = sin x + x cos x |
| Quotient Rule | d/dx(f/g) = (f'g - fg') / g^2 | d/dx(x / (x + 1)) = ((1)(x + 1) - x(1)) / (x + 1)^2 |
| Chain Rule | d/dx(f(g(x))) = f'(g(x))g'(x) | d/dx(sin(x^2)) = 2x cos(x^2) |
| Exponential Rule | d/dx(e^x) = e^x | d/dx(e^x) = e^x |
| Natural Log Rule | d/dx(ln x) = 1/x | d/dx(ln(x)) = 1/x |
Common Derivatives Table
| Function | Derivative |
|---|---|
| x | 1 |
| x^2 | 2x |
| x^3 | 3x^2 |
| x^n | nx^(n-1) |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec^2(x) |
| e^x | e^x |
| ln(x) | 1/x |
| sqrt(x) | 1/(2sqrt(x)) |
| 1/x | -1/x^2 |
First Derivative vs Second Derivative
The first derivative shows the rate of change or slope of a function. If the function is a curve, the first derivative tells you the slope of the tangent line at a point. The second derivative shows how the slope is changing. It can help identify concavity, acceleration, and possible inflection points.
What Is the Chain Rule?
The chain rule is used when one function is inside another function. For sin(x^2), the outside function is sin(u), and the inside function is x^2. The derivative is cos(x^2) times 2x, so the answer is 2x cos(x^2).
What Is the Product Rule?
The product rule is used when two functions are multiplied. If h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x). This rule is common in AP Calculus and college calculus when expressions contain multiplication of two variable-based functions.
What Is the Quotient Rule?
The quotient rule is used when one function is divided by another function. If h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x)) / g(x)^2. This rule is useful for rational expressions and fractions involving functions.
Real-Life Uses of Derivatives
Derivatives are used in many real-life fields. In physics, derivatives help calculate velocity and acceleration. In business, they help analyze marginal cost and marginal revenue. In engineering, they support optimization and design. In medicine, derivatives can describe growth rates and change over time. In economics, they help with supply, demand, and optimization. In data science, derivatives are used in machine learning optimization.
Tips for Entering Functions
- Use x^2 for x squared.
- Use x^3 for x cubed.
- Use sin(x), cos(x), and tan(x) for trigonometric functions.
- Use e^x for exponential functions.
- Use ln(x) for natural logarithm.
- Use sqrt(x) for square root.
- Use 1/x for reciprocal functions.
Important Note About Supported Expressions
This calculus derivative calculator is built for common student-level derivative practice. It supports many common expressions such as polynomials, sin(x), cos(x), tan(x), e^x, ln(x), sqrt(x), 1/x, and simple chain rule examples such as sin(x^2). It is not intended to replace advanced computer algebra systems for very complex symbolic expressions.
Derivative Calculator FAQs
What is a derivative calculator?
A derivative calculator is an online tool that helps you find the derivative of a function. It can show the answer and step-by-step explanation for common calculus problems.
Can this derivative calculator show steps?
Yes, this calculator explains the basic steps and derivative rules used to solve the function.
What is the derivative of x^2?
The derivative of x^2 is 2x using the power rule.
What is the derivative of sin(x)?
The derivative of sin(x) is cos(x).
What is the derivative of e^x?
The derivative of e^x is e^x.
Can I calculate the second derivative?
Yes, you can select the second derivative option to differentiate the function twice.
What is the chain rule?
The chain rule is used when one function is inside another function, such as sin(x^2).
Who can use this derivative calculator?
Students, teachers, tutors, AP Calculus learners, college students, and anyone studying calculus can use this tool.
