Solution
Step 1
Write the function.
[tex]f(x)\text{ = sin\lparen x}^5)[/tex]
Step 2
Use the chain rule to find f'(x)
[tex]\begin{gathered} f^{\prime}(x)\text{ = }\frac{df}{du}\times\frac{du}{dx} \\ \\ u\text{ = x}^5 \\ \\ \frac{du}{dx}\text{ = 5x}^4 \\ f(x)\text{ = sinu} \\ \\ \frac{df}{du}\text{ = cosu} \end{gathered}[/tex]
Step 3
[tex]\begin{gathered} f^{\prime}(x)\text{ = 5x}^4\text{ }\times\text{ cosu} \\ \\ f^{\prime}(x)\text{ = 5x}^4cos(x^5) \end{gathered}[/tex]
Step 4
Substitute x = 4 to find f'(4).
[tex]\begin{gathered} f^{\prime}(4)\text{ = 5}\times4^4\times cos(4^5) \\ \\ f^{\prime}(4)=\text{ 1280}\times cos1024 \\ \\ f^{\prime}(x)\text{ = 715.8} \end{gathered}[/tex]
Final answer
