Derivative Calculator

The derivative of a function f at a point x is the limit f′(x) = lim(h→0) [f(x + h) − f(x)] / h. When it exists, it is the instantaneous rate of change of f, and the slope of the tangent line to the graph at that point. The power rule gives the derivative of a single power term: if f(x) = a·xⁿ, then f′(x) = n·a·xⁿ⁻¹. A constant term (n = 0) has derivative 0, and a·x (n = 1) has derivative a. A sum of power terms is differentiated term by term because differentiation is linear: the derivative of f + g is f′ + g′ (the sum rule), and the derivative of c·f is c·f′ (the constant multiple rule). So for f(x) = 3x² + 5x − 7 the three terms give 6x, 5 and 0, and f′(x) = 6x + 5. Like terms are combined before differentiating, so 3x² + 2x² is treated as 5x², whose derivative is 10x, and terms that cancel, such as −x³ + x³, contribute nothing. For a whole-number exponent the rule follows from the binomial theorem: (x + h)ⁿ = xⁿ + n·xⁿ⁻¹·h + (terms in h² and higher powers of h), so [(x + h)ⁿ − xⁿ] / h tends to n·xⁿ⁻¹ as h → 0. For any real exponent and x > 0, writing xⁿ = e^(n·ln x) and differentiating gives the same result, n·xⁿ⁻¹. That is why the rule also covers x^(1/2) = √x, whose derivative is (1/2)x^(−1/2) = 1/(2√x), and x^−1 = 1/x, whose derivative is −x^−2 = −1/x². Where the result is defined depends on the exponents. A term with a whole-number exponent of 0 or more is defined for every x. A negative exponent divides by a power of x, so it is undefined at x = 0. For a non-integer exponent this calculator uses the usual real-valued convention xⁿ = e^(n·ln x) for x > 0, with 0ⁿ = 0 when n > 0, and treats such a term as undefined for x < 0, even an odd root such as x^(1/3), which does have a real value there. So when x₀ is negative and some exponent is not a whole number, or x₀ is 0 and some exponent in f or f′ is negative, it reports that f′ is undefined at this x. For example, x^(1/2) is 0 at x = 0, but its derivative 0.5x^-0.5 is not defined there: the graph has a vertical tangent. The tangent line at x₀ passes through the point (x₀, f(x₀)) with slope m = f′(x₀). In point-slope form it is y − f(x₀) = m(x − x₀), which the calculator writes in slope-intercept form y = m·x + b with b = f(x₀) − m·x₀. For 3x² + 5x − 7 at x₀ = 2, f(2) = 15 and f′(2) = 17, so b = 15 − 34 = −19 and the tangent line is y = 17x − 19. Near x₀ the tangent line is the best straight-line approximation to f, which is the idea behind linear approximation and Newton's method. Other functions need other rules. A product such as x·(x + 2) needs the product rule (or expanding first), a quotient needs the quotient rule, a composite such as (3x + 1)² needs the chain rule, and sin x, eˣ, aˣ and ln x have derivatives of their own (cos x, eˣ, aˣ·ln a and 1/x). This calculator applies none of them: it refuses such input with a message rather than return a wrong answer. Where a function can be rewritten as a sum of power terms, by expanding products and powers of sums, it can then be entered here. Coefficients in f′(x) and in the tangent line are shown to 10 significant digits, and an exponent that is not a short decimal is shown as a fraction, such as (-2/3). The value f(x₀) and the slope f′(x₀) are rounded to 6 decimal places, and shown in scientific notation from 1,000,000 up and below 0.001.