Question

Let F(x) = f(x ^ 5) * ndG(x) = (f(x)) ^ 5 You also know that a ^ 4 = 3 , f(a) = 2 , f^ prime (a)=13 , f^ prime (a^ 5 )=2

Answer 1

Given the functions:

`\begin{gathered} F(x)=f(x^5) \\ \\ G(x)=(f(x))^5 \end{gathered}`

Additionally, we know that:

`\begin{gathered} a^4=3 \\ f(a)=2 \\ f^(\prime)(a)=13 \\ f^(\prime)(a^5)=2 \end{gathered}`

We use the chain rule to find the derivatives of F(x) and G(x):

`\begin{gathered} F^(\prime)(x)=f^(\prime)(x^5)\cdot(x^5)^(\prime)=5x^4\cdot f^(\prime)(x^5) \\ \\ G^(\prime)(x)=5(f(x))^4\cdot f^(\prime)(x) \end{gathered}`

Finally, evaluating for x = a and using the results stated before:

`\begin{gathered} F^(\prime)(a)=5\cdot a^4\cdot f^(\prime)(a^5)=5\cdot3\cdot2 \\ \\ \therefore F^(\prime)(a)=30 \end{gathered}`
`\begin{gathered} G^(\prime)(a)=5\cdot(f(a))^4\cdot f^(\prime)(a)=5\cdot2^4\cdot13 \\ \\ \therefore G^(\prime)(a)=1040 \end{gathered}`