Question

let ( f(x) = \frac{1}{(x^5 - 7x + 6)^5} ). find ( f(x) ).

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\frac{1}{(x^{5}-7x + 6)^{5}}\) as \( f(x)=(x^{5}-7x + 6)^{-5}\).

Step2: Apply the chain rule

The chain rule states that if \( y = u^{n}\) and \( u = g(x)\), then \( y^\prime=n\cdot u^{n - 1}\cdot u^\prime\). Here, \( n=- 5\) and \( u=x^{5}-7x + 6\).
First, find the derivative of \( u\) with respect to \( x\): \( u^\prime=\frac{d}{dx}(x^{5}-7x + 6)=5x^{4}-7\).
Then, find the derivative of \( y\) with respect to \( u\): \( y^\prime=-5u^{-6}\).

Step3: Combine using the chain rule

Substitute \( u = x^{5}-7x + 6\) back into the derivative. So \( f^\prime(x)=-5(x^{5}-7x + 6)^{-6}\cdot(5x^{4}-7)\).
Rewrite the negative exponent as a fraction: \( f^\prime(x)=-\frac{5(5x^{4}-7)}{(x^{5}-7x + 6)^{6}}\).

Answer:

\(f^\prime(x)=-\frac{5(5x^{4}-7)}{(x^{5}-7x + 6)^{6}}\)