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$^{6}$x u = (type an expression using x as the variable.)

Explanation:

Step1: Define u and y in terms of u

Let \(u = \tan x\), then \(y = u^{6}\).

Step2: Find \(\frac{dy}{du}\)

Using the power - rule \(\frac{d}{du}(u^{n})=nu^{n - 1}\), for \(y = u^{6}\), we have \(\frac{dy}{du}=6u^{5}\).

Step3: Find \(\frac{du}{dx}\)

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

Step4: Apply the chain - rule

The chain - rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). Substituting \(\frac{dy}{du}=6u^{5}\) and \(\frac{du}{dx}=\sec^{2}x\) and \(u = \tan x\) back in, we get \(\frac{dy}{dx}=6\tan^{5}x\cdot\sec^{2}x\).

Answer:

\(u=\tan x\), \(\frac{dy}{dx}=6\tan^{5}x\sec^{2}x\)