Question

let ( f(x) = \frac{1}{(x^5 - 7x + 6)^5} ). find ( f(x) ). submit answer next item practice similar submit answer

Explanation:

Step1: Identify the function type

The function \( f(x)=\frac{1}{(x^{5}-7x + 6)^{5}}\) can be rewritten as \( f(x)=(x^{5}-7x + 6)^{-5}\). We will use the chain rule to differentiate this function. The chain rule states that if \( y = u^{n}\) where \( u\) is a function of \( x\), then \( y'=n\cdot u^{n - 1}\cdot u'\).

Step2: Apply the chain rule

Let \( u=x^{5}-7x + 6\) and \( n=- 5\). First, find the derivative of \( u\) with respect to \( x\): \( u'=\frac{d}{dx}(x^{5}-7x + 6)=5x^{4}-7\).

Then, using the chain rule, the derivative of \( f(x)\) with respect to \( x\) is:
\( f'(x)=n\cdot u^{n - 1}\cdot u'=-5\cdot(x^{5}-7x + 6)^{-5-1}\cdot(5x^{4}-7)\)

Step3: Simplify the expression

Simplify the exponents and the negative exponent:
\( f'(x)=-5\cdot(5x^{4}-7)\cdot(x^{5}-7x + 6)^{-6}=\frac{-5(5x^{4}-7)}{(x^{5}-7x + 6)^{6}}\)

Answer:

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