Question

If f(x) = sin(x ^ 5) , find f^ prime (x)

Answer 1

Step 1

Write the function.

`f(x)\text{ = sin\lparen x}^5)`

Step 2

Use the chain rule to find f'(x)

`\begin{gathered} f^(\prime)(x)\text{ = }(df)/(du)*(du)/(dx) \\ \\ u\text{ = x}^5 \\ \\ (du)/(dx)\text{ = 5x}^4 \\ f(x)\text{ = sinu} \\ \\ (df)/(du)\text{ = cosu} \end{gathered}`

Step 3

`\begin{gathered} f^(\prime)(x)\text{ = 5x}^4\text{ }*\text{ cosu} \\ \\ f^(\prime)(x)\text{ = 5x}^4cos(x^5) \end{gathered}`

Step 4

Substitute x = 4 to find f'(4).

`\begin{gathered} f^(\prime)(4)\text{ = 5}*4^4* cos(4^5) \\ \\ f^(\prime)(4)=\text{ 1280}* cos1024 \\ \\ f^(\prime)(x)\text{ = 715.8} \end{gathered}`

Final answer