Given \( f(x)=x^{5} \) and \( g(x)=(2 x+3) \), find a. the composite function \( h(x)=(f \circ g)(x) \) b. the derivative of \( h \) (hint: use chain) Blank \# \( 1 \square \) A/ Blank \# 2 A/

To solve the problem, we will follow the steps outlined: ### Part a: Finding the Composite Function \( h(x) = (f \circ g)(x) \) 1. **Identify the functions**: - \( f(x) = x^5 \) - \( g(x) = 2x + 3 \) 2. **Compose the functions**: The composite function \( h(x) \) is defined as: \[ h(x) = f(g(x)) \] This means we will substitute \( g(x) \) into \( f(x) \). 3. **Substituting \( g(x) \) into \( f(x) \)**: \[ h(x) = f(g(x)) = f(2x + 3) = (2x + 3)^5 \] Thus, the composite function is: \[ h(x) = (2x + 3)^5 \] ### Part b: Finding the Derivative of \( h \) To find the derivative \( h'(x) \), we will use the chain rule. The chain rule states that if you have a composite function \( h(x) = f(g(x)) \), then the derivative is given by: \[ h'(x) = f'(g(x)) \cdot g'(x) \] 1. **Find \( f'(x) \)**: \[ f'(x) = 5x^4 \] 2. **Find \( g'(x) \)**: \[ g'(x) = 2 \] 3. **Evaluate \( f'(g(x)) \)**: Substitute \( g(x) = 2x + 3 \) into \( f'(x) \): \[ f'(g(x)) = f'(2x + 3) = 5(2x + 3)^4 \] 4. **Apply the chain rule**: Now, we can find \( h'(x) \): \[ h'(x) = f'(g(x)) \cdot g'(x) = 5(2x + 3)^4 \cdot 2 \] \[ h'(x) = 10(2x + 3)^4 \] ### Final Answers - Blank # 1: \( (2x + 3)^5 \) - Blank # 2: \( 10(2x + 3)^4 \)