Derivative

$$f'(x)=0$$

$$f(x)$$	$$f'(x)$$
$$k$$	$$0$$
$$ax$$	$$a$$
$$x^n$$	$$nx^{n-1}$$
$$x^{-n}$$	$$-nx^{-n-1}$$
$$a^x,\, a > 0,\, a \neq 1$$	$$a^x\ln{(a)}$$
$$\cos{(x)}$$	$$-\sin{(x)}$$
$$\sin{(x)}$$	$$\cos{(x)}$$
$$\tan{(x)}$$	$$\sec^2{(x)}$$
$$e^x$$	$$e^x$$
$$\ln{(x)}$$	$$\frac{1}{x}$$
$$\sqrt{x}$$	$$\frac{1}{2\sqrt{x}}$$
$$\frac{1}{x}$$	$$-\frac{1}{x^2}$$

In mathematics, a derivative measures how a function's output changes in response to small changes in its input. It represents the rate of change or slope of the function at a specific point. Denoted as $f'(x)$ or $\frac{dy}{dx}$, it provides crucial information about the function's instantaneous rate of change, enabling us to analyze functions, find tangent lines, and solve optimization problems. Essentially, derivatives help us understand how a function behaves locally, making them a fundamental concept in calculus.