Questions: If f(x)=3 sqrt(x+2) sin (πx+1), find f'(x). DO NOT simplify.

Transcript text: If $f(x)=3 \sqrt{x+2} \sin (\pi x+1)$, find $f^{\prime}(x)$. DO NOT simplify.

Solution

Solution Steps

To find the derivative $ f'(x) $ of the function $ f(x) = 3 \sqrt{x+2} \sin (\pi x + 1) $, we will use the product rule and the chain rule. The product rule states that if you have a function $ f(x) = u(x) \cdot v(x) $, then $ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) $. Here, $ u(x) = 3 \sqrt{x+2} $ and $ v(x) = \sin (\pi x + 1) $. We will also need to use the chain rule to differentiate $ u(x) $ and $ v(x) $.

Step 1: Define the Function

Given the function $ f(x) = 3 \sqrt{x+2} \sin (\pi x + 1) $.

Step 2: Apply the Product Rule

To find the derivative $ f'(x) $, we use the product rule: \[ f'(x) = u'(x) v(x) + u(x) v'(x) \] where $ u(x) = 3 \sqrt{x+2} $ and $ v(x) = \sin (\pi x + 1) $.

Step 3: Differentiate $ u(x) $

\[ u(x) = 3 \sqrt{x+2} \] \[ u'(x) = 3 \cdot \frac{1}{2} (x+2)^{-\frac{1}{2}} \cdot 1 = \frac{3}{2 \sqrt{x+2}} \]

Step 4: Differentiate $ v(x) $

\[ v(x) = \sin (\pi x + 1) \] \[ v'(x) = \cos (\pi x + 1) \cdot \pi \]

Step 5: Combine Using the Product Rule

\[ f'(x) = \left( \frac{3}{2 \sqrt{x+2}} \right) \sin (\pi x + 1) + 3 \sqrt{x+2} \cdot \pi \cos (\pi x + 1) \]

Final Answer

\[ f'(x) = 3 \pi \sqrt{x+2} \cos (\pi x + 1) + \frac{3 \sin (\pi x + 1)}{2 \sqrt{x+2}} \]

$\boxed{f'(x) = 3 \pi \sqrt{x+2} \cos (\pi x + 1) + \frac{3 \sin (\pi x + 1)}{2 \sqrt{x+2}}}$

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