A composite exponential
- Input
- d/dx exp(2x)
- Expected output
- 2·exp(2x)
The derivative of exp(u) is exp(u)·u', and here u = 2x has derivative 2.
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chain rule derivative
The chain rule differentiates a composite function, a function inside another one, by multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function. This page already opens with sin(x²), one of the most common examples in calculus textbooks.
The derivative of exp(u) is exp(u)·u', and here u = 2x has derivative 2.
The derivative of ln(u) is u'/u, and here u = x² + 1 has derivative 2x.
Polynomials, trigonometric functions (sine, cosine, tangent and their inverses), hyperbolic functions, exponentials and logarithms, including combinations built from sums, subtraction, products, quotients and composition (the chain rule). Factorial, the remainder operator (%) and min/max are not elementarily differentiable, and the calculator says so instead of risking a wrong result.
Whenever the argument of a function (inside the parentheses of sin, cos, ln, exp and so on) is an expression other than the bare variable, like x² instead of x. If the argument is just the variable, the chain rule still technically applies, but the argument's derivative is 1 and disappears from the calculation.
The step shows the already combined result (outer derivative times inner derivative) in a single named step, for example "Derivative of sine (chain rule)", and the following steps differentiate the inner expression in their own steps.
Examples by rule
cos(x^2) * (2 * x)
\cos\left(x^{2}\right) \cdot \left(2 \cdot x\right)
No formula renderer on this page: copy the LaTeX above and paste it into an editor that displays it, like a document or a note.
Derivative of sine (chain rule): the derivative of sin(u) is cos(u) * u'.
Power rule (with the chain rule): the derivative of u^n, with n constant, is n * u^(n-1) * u'.
Derivative of the variable with respect to itself: the result is 1.
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