2.2 · Power, exponential, logarithmic, and trigonometric rules

Differentiate term by term

The power rule gives $\displaystyle \frac{d}{dx}x^n=nx^{n-1}$ wherever the function and derivative are defined. Constants differentiate to $\displaystyle 0$, and constant multiples can stay outside. Rewrite roots and reciprocals as powers when that clarifies the rule.

For $\displaystyle 3x^4-2x+7$, differentiate each term to obtain $\displaystyle 12x^3-2$. The exponent becomes a coefficient and decreases by one; the constant does not survive.

$$\frac{d}{dx}(cx^n)=cnx^{n-1}$$

Use the standard function derivatives

The exponential function satisfies $\displaystyle \frac{d}{dx}e^x=e^x$. For $\displaystyle a>0$, $\displaystyle \frac{d}{dx}a^x=a^x\ln a$. The logarithmic rule is $\displaystyle \frac{d}{dx}\ln x=\frac{1}{x}$ for $\displaystyle x>0$; more generally $\displaystyle \ln|x|$ has derivative $\displaystyle 1/x$ for $\displaystyle x\ne0$.

With angles in radians, the derivatives of $\displaystyle \sin x$, $\displaystyle \cos x$, and $\displaystyle \tan x$ are $\displaystyle \cos x$, $\displaystyle -\sin x$, and $\displaystyle \sec^2x$. The remaining standard rules are $\displaystyle (\cot x)'=-\csc^2x$, $\displaystyle (\sec x)'=\sec x\tan x$, and $\displaystyle (\csc x)'=-\csc x\cot x$.

Read a second derivative

For position $\displaystyle s(t)$, velocity is $\displaystyle s'(t)$ and acceleration is $\displaystyle s''(t)$. Acceleration measures how velocity changes, so its units are distance per time squared.

Keep the function domains visible. The derivative of $\displaystyle \sqrt{x}$ is $\displaystyle 1/(2\sqrt{x})$ for $\displaystyle x>0$; the function exists at $\displaystyle 0$ but has no finite derivative there.

$$v(t)=s'(t),\qquad a(t)=s''(t)$$