find the derivative of ( f(x) ).\n( f(x)=cot (5 x-6) )\n( 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))