If y = (2x + 2)$^3$, find $\frac{\delta y}{\delta x}$

To solve for the derivative of $y = (2x + 2)^3$ with respect to $x$, we use the Chain Rule in calculus.

1. Analyze the question

We are given a composite function $y = u^3$, where $u = 2x + 2$. We need to find $\frac{dy}{dx}$.

2. Explain the concept

The Chain Rule states that if $y = f(u)$ and $u = g(x)$, then:

In simpler terms, you differentiate the "outside" function while keeping the "inside" the same, and then multiply it by the derivative of the "inside" function.

3. Show the solution

Step 1: Identify the parts

• Let $u = 2x + 2$
• Then $y = u^3$

Step 2: Differentiate the parts

• Differentiate $y$ with respect to $u$: $\frac{dy}{du} = 3u^2$
• Differentiate $u$ with respect to $x$: $\frac{du}{dx} = 2$ (since the derivative of $2x$ is $2$ and the derivative of $2$ is $0$)

Step 3: Apply the Chain Rule

Step 4: Substitute $u$ back into the equation Since $u = 2x + 2$:

4. Identify the answer

Comparing our result to the given options: A. 3(2x + 2)² B. 6(2x + 2) C. 3(2x + 2) D. 6(2x + 2)²

The correct option is D.

5. Tip

When you see a function raised to a power, like $(ax + b)^n$, always remember the "Power-Chain" shortcut: Bring the power down, subtract one from the power, and multiply by the derivative of the inside.

• Bring down $3 \rightarrow 3(2x + 2)^2$
• Multiply by derivative of $(2x+2)$, which is $2 \rightarrow 3 \times 2(2x + 2)^2 = 6(2x + 2)^2$.