Derivative Calculator

What a derivative tells you

The derivative of a function measures how fast its output changes as the input changes. Geometrically, the value of f′(x) at a point is the slope of the tangent line to the graph there; physically, it is a rate of change such as velocity as the derivative of position. This calculator finds the derivative symbolically, meaning it returns a formula rather than a numerical approximation. Enter a function such as x^3 * sin(x), choose the variable and the order, and the tool applies the sum, product, quotient and chain rules and then simplifies the outcome so you can read it easily.

Higher-order derivatives are obtained by differentiating repeatedly. The second derivative f″ describes curvature: where it is positive the graph bends upward, and where it is negative it bends downward. The third derivative f‴ measures how that curvature itself changes. Select order 1, 2 or 3 and the tool lists every intermediate derivative, so you can check each step of a homework problem or a derivation.

Evaluating and plotting

Type a value in the evaluation field to compute the derivative at a specific point. Constants such as pi and e are accepted, so an entry like pi/2 works. The result is the slope of the original curve when the order is 1, which is useful for finding tangent lines, checking critical points where the derivative is zero, or confirming where a function is increasing. If the derivative does not exist at your point, for instance the square root at zero, the tool says so instead of returning a misleading number.

The canvas draws the original function and the selected derivative over the range you choose. Seeing both curves together builds intuition: peaks and valleys of f line up with zero crossings of f′, and steep sections of f correspond to large values of f′. The vertical scale adjusts automatically, and sudden jumps near asymptotes are not joined by a false line.

Tips for writing expressions

Use ^ for powers, * for multiplication (a number directly before a letter, as in 3x, is also understood) and parentheses to group terms. Functions are written in lowercase, for example sin(x), exp(x), log(x) for the natural logarithm and sqrt(x). Symbols other than the chosen variable are treated as constants, which makes partial derivatives easy: differentiate with respect to y while x stays fixed. If the expression cannot be read, you will get a plain message describing what went wrong. Everything runs in your browser, so your work stays private.