Derivative Calculator
with Step-by-Step
Explanations
Enter any function — Math GPT identifies the rule, shows each step, and explains why, not just what. From power rule basics to implicit differentiation.
Differentiation Rules
Every Rule. Fully Explained.
Math GPT identifies which rule applies to your function — and shows exactly how it works, step by step.
Power Rule
The simplest and most frequently used rule. Applies whenever the variable has an exponent.
Chain Rule
Used for composite functions — a function inside another function.
Product Rule
Applies when two functions are multiplied. Math GPT identifies both factors automatically.
Quotient Rule
For functions expressed as a fraction. Frequently tested in Calculus I exams.
Trigonometric
Sine, cosine, tangent and their inverses — each with full derivation shown.
Exponential & Logarithmic
eˣ, aˣ, ln(x), logₐ(x) — all standard forms covered with clear reasoning.
In Action
Derivative Calculator — Worked Examples
Three real problems showing exactly how Math GPT annotates each step.
2nd Derivative
Second Derivative Calculator
The second derivative f”(x) measures how the rate of change itself is changing. It determines whether a function is concave up or concave down — and locates inflection points.
Toggle “2nd derivative” above the input to have Math GPT differentiate twice, with each step annotated independently.
- ✓ Identify concavity intervals
- ✓ Find inflection points
- ✓ Verify local min/max with the Second Derivative Test
Multivariable
Partial Derivative Calculator
When a function has more than one variable, partial derivatives measure how the function changes with respect to each variable independently.
- ✓ Functions of 2 or 3 variables
- ✓ Gradient and directional derivatives
- ✓ Used in Calculus III, multivariable analysis, ML
Who Uses This Derivative Calculator
High School Students
AP Calculus AB/BC preparation. Annotated steps teach the logical sequence that exam graders reward.
College Students
Calculus I and II homework verification. Identify exactly where your working diverges from the correct method.
Engineers & Researchers
Fast symbolic verification of applied derivatives before committing to a report.
Teachers & Tutors
Generate worked examples instantly. Demonstrate differentiation live in class.
What Is a Derivative — And Why the Steps Matter
A derivative measures the instantaneous rate of change of a function at any given point. If f(x) describes position, f'(x) describes velocity. Derivatives appear in physics, economics, machine learning, engineering, and every quantitative field.
The problem with most online math solvers is that they return a derivative without explaining what rule was applied or why. Math GPT takes a different approach: every step carries a plain-language annotation so the method becomes as legible as the result.
How to Use This Derivative Calculator with Steps
Enter your function in standard mathematical notation. Math GPT accepts polynomial expressions, trigonometric functions, exponentials, logarithms, and combinations of all of the above.
- Select the derivative order. Toggle between first and second derivative using the buttons above the input.
- Review the annotated steps. Each step names the rule applied and explains the reasoning before showing the calculation.
- Use the worked examples. Click “Try in solver” on any example to load it into the calculator.
- Copy the LaTeX output directly into Overleaf, Word, or any academic editor.
Derivative Calculator with Steps vs Symbolab vs Wolfram Alpha
Symbolab and Wolfram Alpha are powerful tools, but they optimize for the result. Steps are often hidden behind a paywall or presented without explanation. Math GPT shows the reasoning at each step in plain English — transforming a calculator from an answer generator into an educational tool that outperforms general AI.
Derivative Rules Covered
Chain Rule Derivative Calculator
The chain rule is the single most tested differentiation concept in AP Calculus and Calculus I. It applies whenever a function is nested inside another — sin(x²), e^(3x), ln(x²+1). Math GPT identifies the outer and inner functions automatically and applies d/dx[f(g(x))] = f'(g(x))·g'(x) with full annotation.
Product Rule and Quotient Rule
When two functions are multiplied or divided, the standard power rule no longer applies directly. Math GPT recognizes both patterns and shows the complete derivation including any simplification at the end.
Implicit Differentiation
When a relationship between x and y cannot be written as an explicit function, implicit differentiation differentiates both sides with respect to x. Enter the equation and specify dy/dx to get a full annotated solution.
Second Derivative and Concavity
The second derivative f”(x) determines concavity: f”(x) > 0 means concave up; f”(x) < 0 means concave down. Toggle the derivative order above to have Math GPT compute and annotate each differentiation pass — up to the 5th derivative.
Derivative at a Point Calculator
To evaluate a derivative at a specific value — for example, finding f'(2) for f(x) = x³ — enter the function and fill in the “Evaluate at x =” field above. Math GPT computes the general derivative first, then substitutes the value with full annotation, giving you both f'(x) and the numerical slope at that point.
Derivative from the Limit Definition
For students learning the formal definition, Math GPT can work through the limit process: f'(x) = lim(h→0) [f(x+h) − f(x)] / h — step by step, showing how each term simplifies before taking the limit. Type your function and add “using limit definition” to the input to trigger this approach.
Derivative Calculator — FAQ
A derivative measures how fast a function changes at any given point. If f(x) tells you an object’s position, f'(x) tells you its speed. If f(x) is a revenue function, f'(x) tells you the marginal revenue.
The chain rule applies when you differentiate a composite function. The formula is d/dx[f(g(x))] = f'(g(x)) times g'(x). Differentiate the outer function leaving the inner unchanged, then multiply by the derivative of the inner function. Math GPT identifies the outer and inner functions automatically.
Yes. Enter an equation like x squared plus y squared equals 25 and specify that you need dy/dx. Math GPT differentiates both sides with respect to x and solves for dy/dx with full step annotations.
The first derivative f'(x) gives the rate of change of a function. The second derivative f”(x) tells you how the slope itself is changing, which determines concavity and identifies inflection points.
Yes. Math GPT’s derivative calculator is completely free with no sign up required. Enter any function and get step-by-step annotated results immediately.
Yes. The calculator covers all derivative rules tested in AP Calculus AB and BC, including power rule, chain rule, product and quotient rules, implicit differentiation, and higher-order derivatives.
Math GPT explains why each rule is applied in plain English at every step. Symbolab and Wolfram Alpha often require a paid subscription to see full solutions. Math GPT is free and designed to help you understand the method, not just copy the answer.
Ready to Find Your Derivative?
Enter any function above — or return to Math GPT for the full AI math solver.