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 = sec$^{4}$x u = □ (type an expression using x as the variable.)

Explanation:

Step1: Define u and y

Let $u = \sec x$, then $y = u^{4}$.

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

Using the power - rule for differentiation, if $y = u^{4}$, then $\frac{dy}{du}=4u^{3}$.

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

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

Step4: Use the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=4u^{3}$ and $\frac{du}{dx}=\sec x\tan x$ and $u = \sec x$ into the chain - rule formula, we get $\frac{dy}{dx}=4(\sec x)^{3}\cdot\sec x\tan x$.

Step5: Simplify the result

$\frac{dy}{dx}=4\sec^{4}x\tan x$.

Answer:

$u = \sec x$, $\frac{dy}{dx}=4\sec^{4}x\tan x$