Question

1. f(x)=tan³x

f(x)=3tanxsec²x or f(x)=3tan²xsec²x

Explanation:

Step1: Apply chain - rule

Let $u = \tan x$, then $y = u^{3}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$: if $y = u^{3}$, then $\frac{dy}{du}=3u^{2}$. Second, find $\frac{du}{dx}$: since $u=\tan x$, then $\frac{du}{dx}=\sec^{2}x$.

Step2: Substitute $u$ back

Substitute $u = \tan x$ into $\frac{dy}{du}\cdot\frac{du}{dx}$. We get $\frac{dy}{dx}=3\tan^{2}x\sec^{2}x$.

Answer:

$f^{\prime}(x)=3\tan^{2}x\sec^{2}x$