Given the following functions: \( f(u)=u^{7 / 2} \) and \( g(x)=x^{8}+1 \). Find: \( f(g(x))=\square \) \( f^{\prime}(u)=\square \) \( f^{\prime}(g(x))=\square \) \( g^{\prime}(x)=\square \) Therefore, the derivative of \( f(g(x)) \) is

To solve the problem, we will compute each of the required expressions step by step. 1. **Finding \( f(g(x)) \)**: \[ f(g(x)) = f(x^8 + 1) = (x^8 + 1)^{7/2} \] So, \[ f(g(x)) = (x^8 + 1)^{7/2} \] 2. **Finding \( f'(u) \)**: To find the derivative of \( f(u) = u^{7/2} \), we use the power rule: \[ f'(u) = \frac{7}{2} u^{(7/2) - 1} = \frac{7}{2} u^{5/2} \] 3. **Finding \( f'(g(x)) \)**: We substitute \( g(x) \) into \( f'(u) \): \[ f'(g(x)) = f'(x^8 + 1) = \frac{7}{2} (x^8 + 1)^{5/2} \] 4. **Finding \( g'(x) \)**: To find the derivative of \( g(x) = x^8 + 1 \), we again use the power rule: \[ g'(x) = 8x^{7} \] 5. **Finding the derivative of \( f(g(x)) \)**: We will use the chain rule to find the derivative of \( f(g(x)) \): \[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \] Substituting the expressions we found: \[ \frac{d}{dx} f(g(x)) = \left( \frac{7}{2} (x^8 + 1)^{5/2} \right) \cdot (8x^7) \] Simplifying this gives: \[ \frac{d}{dx} f(g(x)) = 28x^7 (x^8 + 1)^{5/2} \] Now, summarizing the results: - \( f(g(x)) = (x^8 + 1)^{7/2} \) - \( f'(u) = \frac{7}{2} u^{5/2} \) - \( f'(g(x)) = \frac{7}{2} (x^8 + 1)^{5/2} \) - \( g'(x) = 8x^7 \) - The derivative of \( f(g(x)) \) is \( 28x^7 (x^8 + 1)^{5/2} \)