Question

find the derivative of ( f(x) ).
( f(x)=cot (5 x-6) )
( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

Let \(u = 5x-6\), then \(y=\cot(u)\). The chain rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
The derivative of \(\cot(u)\) with respect to \(u\) is \(-\csc^{2}(u)\), and the derivative of \(u = 5x - 6\) with respect to \(x\) is \(5\).

Step2: Substitute back \(u\)

Substitute \(u = 5x-6\) into \(\frac{dy}{dx}\). So \(\frac{dy}{dx}=- 5\csc^{2}(5x - 6)\)

Answer:

\(-5\csc^{2}(5x - 6)\)