Derivative Calculator

What Is a Derivative Calculator?

A derivative measures how a function changes as its input changes — it's the instantaneous rate of change at any point on a curve. Derivatives are foundational to calculus, powering everything from velocity and acceleration in physics to optimization in machine learning and economics. Whether you're finding the slope of a tangent line or maximizing profit, derivatives are the tool you reach for.

Computing derivatives by hand means memorizing dozens of rules — power rule, product rule, quotient rule, chain rule, trigonometric identities — and applying them without a single sign error. Our free derivative calculator gives you symbolic derivatives instantly, shows you exactly which differentiation rules were applied at each step, and plots both f(x) and f'(x) on an interactive graph so you can visualize the relationship between a function and its rate of change.

Common Use Cases

• Polynomial derivatives - find the derivative of expressions like x³ + 4x² - 7x + 2 using the power rule
• Trigonometric functions - differentiate sin(x), cos(x), tan(x), and their compositions
• Exponential and logarithmic functions - compute derivatives of e^x, ln(x), and their variations
• Chain rule applications - handle nested functions like sin(x²) or e^(3x+1) where the chain rule is required
• Product rule expressions - differentiate products like x·sin(x) or x²·e^x where two functions are multiplied
• Quotient rule expressions - find derivatives of fractions like sin(x)/x or (x²+1)/(x-3)

Why Computing Derivatives by Hand Is Challenging

• Too many rules to remember - power rule, product rule, quotient rule, chain rule, and dozens of trigonometric and exponential identities all need to be recalled correctly
• Algebraic simplification is tedious - after applying differentiation rules, the raw result often needs several steps of simplification to reach a clean final answer
• Nested functions compound complexity - expressions like e^(sin(x²)) require applying the chain rule multiple times in succession, and one missed layer ruins the entire result
• Sign errors propagate silently - a single misplaced negative sign early in the process carries through every subsequent step, giving a completely wrong answer that still looks plausible

How It Works

Enter any mathematical expression using standard notation — the calculator accepts variables, constants, arithmetic operators, and common functions like sin, cos, exp, ln, and sqrt. The symbolic computation engine powered by mathjs parses your expression, applies the appropriate differentiation rules, and returns the simplified derivative.

Beyond just the answer, the calculator shows you which rules were used at each step so you can follow the logic. It also plots both the original function f(x) and its derivative f'(x) on an interactive graph, letting you see how the slope of f(x) corresponds to the value of f'(x) at every point.

Example

Input: x^3 + 2*x^2 - 5*x + 3

Output: 3x² + 4x - 5

The power rule is applied to each term: x³ becomes 3x², 2x² becomes 4x, -5x becomes -5, and the constant 3 drops out.

Key Examples

• Power rule - the derivative of x⁵ is 5x⁴, bringing the exponent down and reducing it by one
• Trigonometric derivative - the derivative of sin(x) is cos(x), one of the fundamental trig derivatives
• Chain rule - the derivative of e^(2x) is 2e^(2x), multiplying by the derivative of the inner function
• Product rule - the derivative of x·sin(x) is sin(x) + x·cos(x), applying d(uv) = u'v + uv'
• Quotient rule - the derivative of sin(x)/x uses (x·cos(x) - sin(x))/x², following the low-d-high-minus-high-d-low pattern

Benefits

• Instant symbolic derivatives - get exact answers in symbolic form, not numerical approximations
• See differentiation rules applied - understand which rules (power, chain, product, quotient) were used at each step
• Visualize f(x) and f'(x) together - the interactive graph shows both curves so you can see how slope relates to derivative values
• 100% browser-based - no downloads, no accounts, no software to install — just type and compute
• Perfect for studying calculus - use it to check homework, prepare for exams, or build intuition about how derivatives behave

Frequently Asked Questions

What expressions does the derivative calculator support?

The calculator supports polynomials, trigonometric functions (sin, cos, tan, sec, csc, cot), exponential functions (e^x), natural logarithms (ln), square roots, and any combination of these using addition, subtraction, multiplication, division, and exponentiation. You can enter expressions like x^3 + sin(2*x) - e^x/ln(x).

How does it show the differentiation steps?

After computing the derivative, the calculator displays which rules were applied — such as power rule, chain rule, product rule, or quotient rule — so you can follow the reasoning from the original expression to the final answer.

Is the derivative computed on the server or in my browser?

All computation happens entirely in your browser using the mathjs symbolic math library. Your expressions are never sent to a server, which means instant results and complete privacy.

Can it handle multivariable derivatives or partial derivatives?

The calculator currently computes derivatives with respect to x. For multivariable functions, you can treat other variables as constants and differentiate with respect to x to get the partial derivative ∂f/∂x.

What about higher-order derivatives like f''(x) or f'''(x)?

You can compute higher-order derivatives by taking the output of the first derivative and entering it back into the calculator. For example, differentiate x⁴ to get 4x³, then differentiate 4x³ to get 12x² for the second derivative.

Is the derivative calculator free to use?

Yes, the derivative calculator is completely free with no usage limits. You can compute as many derivatives as you need for homework, studying, or professional work.

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