Questions: Let f(x)=(x^(1 / 3)+5)(3 x^(1 / 2)+7) Find f'(x)= □ Remember: You do not need to simplify your answer.

Solution Steps

To find the derivative \( f'(x) \) of the function \( f(x) = \left(x^{1/3} + 5\right)\left(3x^{1/2} + 7\right) \), we will use the product rule of differentiation. The product rule states that if \( f(x) = g(x) \cdot h(x) \), then \( f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) \). We will first find the derivatives of \( g(x) = x^{1/3} + 5 \) and \( h(x) = 3x^{1/2} + 7 \), and then apply the product rule.

We start with the function \( f(x) = \left(x^{1/3} + 5\right)\left(3x^{1/2} + 7\right) \).

To find the derivative \( f'(x) \), we use the product rule: \[ f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) \] where \( g(x) = x^{1/3} + 5 \) and \( h(x) = 3x^{1/2} + 7 \).

First, we find the derivatives of \( g(x) \) and \( h(x) \): \[ g'(x) = \frac{d}{dx}\left(x^{1/3} + 5\right) = \frac{1}{3}x^{-2/3} \] \[ h'(x) = \frac{d}{dx}\left(3x^{1/2} + 7\right) = \frac{3}{2}x^{-1/2} \]

Substitute \( g'(x) \) and \( h'(x) \) into the product rule formula: \[ f'(x) = \left(\frac{1}{3}x^{-2/3}\right)\left(3x^{1/2} + 7\right) + \left(x^{1/3} + 5\right)\left(\frac{3}{2}x^{-1/2}\right) \]

Combine and simplify the terms: \[ f'(x) = \frac{1}{3} \cdot 3x^{1/2} \cdot x^{-2/3} + \frac{1}{3} \cdot 7x^{-2/3} + \frac{3}{2} \cdot x^{1/3} \cdot x^{-1/2} + \frac{3}{2} \cdot 5x^{-1/2} \] \[ f'(x) = x^{1/2 - 2/3} + \frac{7}{3}x^{-2/3} + \frac{3}{2}x^{1/3 - 1/2} + \frac{15}{2}x^{-1/2} \] \[ f'(x) = x^{-1/6} + \frac{7}{3}x^{-2/3} + \frac{3}{2}x^{-1/6} + \frac{15}{2}x^{-1/2} \]

Final Answer