Derivative Calculator

This calculator evaluates the derivative — the instantaneous slope of the tangent line — of a chosen function family at a specific x Value, using the closed-form derivative formula for whichever function type is selected rather than a numerical approximation. Of the three movable numeric inputs, x Value has the strongest pull on the reported derivative at the calculator's polynomial defaults, ahead of both the Coefficient and the Exponent/Parameter, because the derivative formula raises x Value to a power (Exponent minus 1) while Coefficient and Exponent only ever multiply the result by a flat factor. Function Type itself moves the derivative more than any single numeric input, though — switching between polynomial, trig, exponential, and logarithmic swaps the entire rule being applied, not just one number feeding into it. It also means the sign of that pull is not fixed: a larger x Value raises the derivative for the default polynomial, but for the trig function the same move can lower it, since sin and cos flip sign as their argument sweeps past a multiple of pi — the direction of x Value's effect is a property of which function is selected, not a fact about x Value on its own.

Coefficient and Exponent/Parameter happen to move the result by the identical proportion at the default x Value of 1, since 1 raised to any power is still 1 — the term x Value raised to (Exponent minus 1) collapses to exactly 1 regardless of what the exponent is whenever x Value itself is 1. That's a coincidence specific to that starting point, not a general property of the formula, since raising x Value away from 1 breaks the tie between them immediately. This page performs no domain checking: for the Logarithmic function type, an x Value of exactly 0 or a negative number would normally make the logarithm undefined, but rather than reporting an error, the calculator quietly substitutes the function value 0 and computes the derivative as 0 as well, which can look like a legitimate answer rather than the placeholder it actually is.