write the function in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x. y =…

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.)

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.)

Answer

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)