Question

write the function below in the form y = f(u) and u = g(x), then find $\frac{dy}{dx}$ as a function of x. y = cot(tan x) what are the functions f(u) and g(x)? f(u) = g(x) =

Explanation:

Step1: Identify inner - outer functions

Let \(u = g(x)=\tan x\) and \(y = f(u)=\cot u\).

Step2: Find derivatives of \(f(u)\) and \(g(x)\)

The derivative of \(y = f(u)=\cot u\) with respect to \(u\) is \(f^\prime(u)=-\csc^{2}u\). The derivative of \(u = g(x)=\tan x\) with respect to \(x\) is \(g^\prime(x)=\sec^{2}x\).

Step3: Apply chain - rule

By the chain - rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\), substituting \(f^\prime(u)\) and \(g^\prime(x)\) we get \(\frac{dy}{dx}=-\csc^{2}(\tan x)\cdot\sec^{2}x\).

Answer:

\(f(u)=\cot u\), \(g(x)=\tan x\), \(\frac{dy}{dx}=-\csc^{2}(\tan x)\sec^{2}x\)