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

Answer

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)