Question

let ( f(x)=\frac{x}{x + 7} ). find the values of ( x ) where ( f^{prime}(x)=6 ).
give exact answers (not decimal approximations).
the greater solution is ( x=)
the lesser solution is ( x=)
question help: video message instructor
submit question jump to answer

Explanation:

Step1: Differentiate \( f(x) \) using quotient rule

The quotient rule states that if \( f(x)=\frac{u}{v} \), then \( f^{\prime}(x)=\frac{u^{\prime}v - uv^{\prime}}{v^{2}} \). Here \( u = x\), \(u^{\prime}=1\), \(v=x + 7\), \(v^{\prime}=1\).

$$
LATEXBLOCK0
$$

Step2: Set \( f^{\prime}(x)=6 \) and solve for \( x \)

Set \(\frac{7}{(x + 7)^{2}}=6\). Cross - multiply to get \(7 = 6(x + 7)^{2}\). Then \((x + 7)^{2}=\frac{7}{6}\). Take square roots: \(x+7=\pm\sqrt{\frac{7}{6}}=\pm\frac{\sqrt{42}}{6}\). Solve for \(x\): \(x=-7\pm\frac{\sqrt{42}}{6}\).

Answer:

The greater solution is \(x=-7+\frac{\sqrt{42}}{6}\).
The lesser solution is \(x=-7-\frac{\sqrt{42}}{6}\).