Questions: Find dy/dx when y=(4x+7)^3. 12(4x+7)^2 (4x+7)^4/16 84(4x+7)^2 3u^2 3(4x+7)^2

Transcript text: 2. Find $\frac{d y}{d x}$ when $y=(4 x+7)^{3}$. $12(4 x+7)^{2}$ $\frac{(4 x+7)^{4}}{16}$ $84(4 x+7)^{2}$ $3 u^{2}$ $3(4 x+7)^{2}$

Solution

Step 1: Identify the function to differentiate

The given function is $ y = (4x + 7)^3 $.

Step 2: Apply the chain rule

To find $\frac{dy}{dx}$, use the chain rule. Let $ u = 4x + 7 $, so $ y = u^3 $.

Step 3: Differentiate $ y $ with respect to $ u $

\[ \frac{dy}{du} = 3u^2 \]

Step 4: Differentiate $ u $ with respect to $ x $

\[ \frac{du}{dx} = 4 \]

Step 5: Combine the derivatives using the chain rule

\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3u^2 \cdot 4 = 12u^2 \]

Step 6: Substitute $ u = 4x + 7 $ back into the expression

\[ \frac{dy}{dx} = 12(4x + 7)^2 \]

The correct answer is $ 12(4x + 7)^2 $.

$\boxed{12(4x + 7)^{2}}$

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