Questions: Find G'(w) if G(w) = 7/9 w^(-6) + 7 sqrt(w).

Transcript text: Find $G^{\prime}(w)$ if $G(w)=\frac{7}{9 w^{6}}+7 \sqrt{w}$.

Solution

Solution Steps

Step 1: Define the Function

We start with the function $ G(w) $ given by: \[ G(w) = \frac{7}{9 w^{6}} + 7 \sqrt{w} \]

Step 2: Differentiate the Function

To find the derivative $ G^{\prime}(w) $, we apply the power rule to each term in the function. The derivative of $ \frac{7}{9 w^{6}} $ is calculated as follows: \[ \frac{d}{dw}\left(\frac{7}{9 w^{6}}\right) = -\frac{14}{3 w^{7}} \] The derivative of $ 7 \sqrt{w} $ is: \[ \frac{d}{dw}(7 \sqrt{w}) = 7 \cdot \frac{1}{2\sqrt{w}} = \frac{7}{2\sqrt{w}} \]

Step 3: Combine the Results

Combining the derivatives from both terms, we have: \[ G^{\prime}(w) = -\frac{14}{3 w^{7}} + \frac{7}{2\sqrt{w}} \]