Question

Let F(x) = f(x ^ 7) and G(x) = (f(x)) ^ 7 You also know that , f(a) = 2 , f^ prime (a)=2 , f^ prime (a^ 7 )=13; a ^ 6 = 8

Answer 1

Given

$\begin{gathered} F(x)=f(x^7) \\ G(x)=(f(x))^7 \end{gathered}$

And,

$a^6=8,\text{ }f(a)=2,\text{ }f^{\prime}(a)=2,\text{ }f^{\prime}(a^7)=13$

To find:

The value of F'(a) and G'(a).

Explanation:

It is given that,

$\begin{gathered} F(x)=f(x^7) \\ G(x)=(f(x))^7 \end{gathered}$

And,

$a^6=8,\text{ }f(a)=2,\text{ }f^{\prime}(a)=2,\text{ }f^{\prime}(a^7)=13$

That implies,

$\begin{gathered} F^{\prime}(x)=\frac{d}{dx}(f(x^7)) \\ =f^{\prime}(x^7)\cdot7x^6 \\ F^{\prime}(a)=f^{\prime}(a^7)\cdot7a^6 \\ =13\cdot(7\times8) \\ =13\times56 \\ =728 \end{gathered}$

Also,

$\begin{gathered} G^{\prime}(x)=\frac{d}{dx}((f(x))^7) \\ =7(f(x))^6\cdot f^{\prime}(x) \\ G^{\prime}(a)=7(f(a))^6\cdot f^{\prime}(a) \\ =7\times(2)^6\cdot2 \\ =7\times2^7 \\ =7\times128 \\ =896 \end{gathered}$

Hence, the value of F'(a)=728, and the value of G'(a)=896.