Derivative Calculator

How to calculate a derivative with this solver

A derivative is the instantaneous rate of change of a function. If f tells you a value at x, f′ tells you how fast that value is changing at x. On this page you type the function — x^2, sin(x), ln(x), or something longer — and press Solve. The calculator returns the derivative, names the rule it used, and lists the steps so you can check the work instead of copying a bare answer.

Common functions are differentiated with deterministic calculus rules: power, constant, sum, product, quotient, and chain, plus the standard derivatives of sine, cosine, tangent, exponentials, and logarithms. An explanation in words sits next to those steps. You can change the expression and solve again without reloading the page.

Write powers with ^, and put parentheses around anything that belongs together. sin(2*x) is unambiguous. sin2x is easy for both students and parsers to misread. Functions the solver recognizes include sin, cos, tan, sec, csc, cot, arcsin, arccos, arctan, ln, log, exp, and sqrt.

Derivative rules the solver uses

The power rule is the one you will use most: the derivative of x^n is n·x^(n−1). The derivative of x^2 is 2x. The derivative of x^3 is 3x^2. A constant has derivative 0, so the derivative of pi is 0 even though pi is written as a letter. The sum rule then says you may differentiate one term at a time: the derivative of x^2 + 1 is just 2x.

The product rule is for a product of two functions, (uv)′ = u′v + uv′. The derivative of x·sin(x) is sin(x) + x·cos(x). The quotient rule is for a fraction, (u/v)′ = (u′v − uv′) / v^2. That is why the derivative of 1/x is −1/x^2. Rewrite 1/x^2 as x^(−2) and the power rule gives −2x^(−3), which is −2/x^3. Both routes have to agree; if they do not, a sign was dropped.

The chain rule is for a function inside a function: the derivative of f(g(x)) is f′(g(x))·g′(x). The derivative of sin(2x) is cos(2x) times 2, which is 2 cos(2x). Skipping that inner 2 is the usual wrong answer.

Trigonometric derivatives worth memorizing: the derivative of sin(x) is cos(x), of cos(x) is −sin(x), of tan(x) is sec^2(x), of cot(x) is −csc^2(x), of sec(x) is sec(x)tan(x), and of csc(x) is −csc(x)cot(x). Inverse trig is a different family. The derivative of arcsin(x) is 1/√(1−x^2), of arccos(x) is −1/√(1−x^2), and of arctan(x) is 1/(1+x^2). Searches for sin^−1(x) or tan^−1(x) mean these inverse functions, not 1/sin(x).

Exponentials and logarithms: the derivative of e^x is e^x, of e^(2x) is 2e^(2x), of ln(x) is 1/x, and of ln(2x) is also 1/x because the constant factor 2 disappears into the chain rule. The derivative of x·ln(x) needs the product rule and equals ln(x) + 1.

Mistakes that change the derivative

Dropping the chain-rule factor. The derivative of sin(2x) is not cos(2x). The derivative of the inside, which is 2, has to multiply the outside derivative.

Treating a product as a sum. The derivative of x·sin(x) is not cos(x). Each factor contributes a term, and both terms stay.

Reading inverse trig as a reciprocal. arcsin(x) is not csc(x). The derivative of arcsin(x) is not the derivative of 1/sin(x). If a problem writes sin^−1(x), check whether the course means the inverse function. On these pages, arcsin means the inverse.

Differentiating a constant that looks important. pi, numeric coefficients standing alone, and e^0 all have derivative 0. e^x does not, because the exponent still depends on x.

Mixing up ln and log. Here ln(x) is the natural logarithm, base e, and its derivative is 1/x. If your class uses log for base 10, that derivative has an extra factor of 1/ln(10). Type the base you were given.

Integrals, limits, and the other tools

Differentiation answers “how fast is this changing?” An integral asks for an antiderivative, and a limit asks what the function approaches. Those are separate calculators: the integral calculator, the limit calculator, the matrix calculator for determinants and inverses, and the differential-equation solver for first- and second-order equations. The calculus wiki states the rules above as short articles when you want the definition without running another calculation.

If a result looks wrong, check parentheses and the variable first. The solver differentiates with respect to x. An expression with no x in it is a constant, and the derivative will be 0.

Worked derivatives students look up

• Derivative of sin(2x) — Chain rule: cos(2x) · 2 = 2 cos(2x).
• Derivative of 1/x^2 — Rewrite as x^(−2). Power rule: −2/x^3.
• Derivative of tan(x) — sec^2(x). The quotient rule on sin/cos gives the same result.
• Derivative of arcsin(x) — 1/√(1−x^2), for x between −1 and 1.
• Derivative of cot(x) — −csc^2(x). The minus sign belongs in the answer.
• Derivative of 2x — A straight line of slope 2. The derivative is the constant 2.
• Derivative of pi — Pi does not depend on x, so the derivative is 0.
• Derivative of x ln(x) — Product rule: ln(x) + 1.

Frequently asked questions