Enter any function into the Derivative Calculator and choose your differentiation variable and derivative order (1st through 5th). You get back the derivative result along with a breakdown of which rules were applied — power rule, chain rule, product rule, and more. Supports polynomials, trigonometric, exponential, and logarithmic functions.
Derivative Result
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Derivative Expression
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Value at Given Point
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Rules Applied
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Derivative Order
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A derivative measures how a function changes as its input changes — it represents the instantaneous rate of change or the slope of the tangent line at any point on the curve. Formally, the derivative f'(x) is defined as the limit of the difference quotient as the interval approaches zero. Derivatives are foundational to calculus and are used in physics, engineering, economics, and many other fields.
The calculator supports polynomials (e.g. x³ + 2x), trigonometric functions (sin, cos, tan), inverse trig functions (arcsin, arctan), exponential functions (e^x, a^x), logarithms (ln(x), log(x)), and composite or mixed expressions. Use standard notation: * for multiplication, ^ for exponentiation, and parentheses to clarify order of operations.
Write the function using standard math notation. For example: x^3 + 2*x - 1 for a polynomial, sin(x)*cos(x) for a trig product, e^(x^2) for a composite exponential, or ln(x^2 + 1) for a logarithm. Always use parentheses around function arguments and around exponents that contain expressions.
The most important rules are: the Power Rule (d/dx[xⁿ] = n·xⁿ⁻¹), the Product Rule (d/dx[f·g] = f'g + fg'), the Quotient Rule (d/dx[f/g] = (f'g − fg')/g²), the Chain Rule (d/dx[f(g(x))] = f'(g(x))·g'(x)), and standard derivatives for sin, cos, eˣ, and ln(x). The calculator identifies and applies these rules automatically.
A second derivative f''(x) is the derivative of the first derivative — it measures the rate of change of the slope, which relates to concavity and acceleration. Higher-order derivatives (3rd, 4th, 5th) are obtained by differentiating repeatedly. They appear in Taylor series expansions, jerk (rate of change of acceleration), and advanced physics models.
Yes. Enter a numeric value in the 'Evaluate at Point' field and the calculator will substitute that value into the derivative expression to give you f'(a). For example, if f(x) = x² and you evaluate at x = 3, the result is f'(3) = 6. This is useful for finding the exact slope of a tangent line at a given x.
Implicit differentiation is used when y is not explicitly defined as a function of x — for example in equations like x² + y² = 25. You differentiate both sides with respect to x and treat y as a function of x, applying the chain rule wherever y appears. This technique is essential for curves like circles, ellipses, and other relations.
Derivatives have wide real-world applications: in physics they describe velocity and acceleration; in economics they find marginal cost and revenue; in engineering they model rates of heat transfer or current flow; in machine learning, gradient descent uses derivatives to minimize loss functions. Anywhere a rate of change matters, derivatives are the tool of choice.