Question

write the function in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x. y = tan$^{8}$x u = (type an expression using x as the variable.)

Explanation:

Step1: Decompose the function

Let $u = \tan x$, then $y = u^{8}$.

Step2: Find $\frac{dy}{du}$

Using the power - rule, if $y = u^{8}$, then $\frac{dy}{du}=8u^{7}$.

Step3: Find $\frac{du}{dx}$

The derivative of $u=\tan x$ with respect to $x$ is $\frac{du}{dx}=\sec^{2}x$.

Step4: Use the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=8u^{7}$ and $\frac{du}{dx}=\sec^{2}x$ and $u = \tan x$ back in, we get $\frac{dy}{dx}=8\tan^{7}x\cdot\sec^{2}x$.

Answer:

$u=\tan x$, $\frac{dy}{dx}=8\tan^{7}x\sec^{2}x$