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 = tan x (type an expression using x as the variable.) y = (type an expression using u as the variable.)

Explanation:

Step1: Define y in terms of u

Given \(u = \tan x\), and \(y=\tan^{6}x\), we can write \(y = u^{6}\).

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

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

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

Since \(u=\tan x\), and \(\frac{d}{dx}(\tan x)=\sec^{2}x\), so \(\frac{du}{dx}=\sec^{2}x\).

Step4: Use the chain - rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)

Substitute \(\frac{dy}{du}=6u^{5}\) and \(\frac{du}{dx}=\sec^{2}x\) into the chain - rule. Replace \(u\) with \(\tan x\), we get \(\frac{dy}{dx}=6(\tan x)^{5}\cdot\sec^{2}x = 6\tan^{5}x\sec^{2}x\).

Answer:

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