Question

Using chain rule:
y=3(2^2x)

Answer 1

To differentiate the function \( y = 3(2^{2x}) \) using the chain rule, we start by identifying the outer and inner functions.

Let \( u = 2^{2x} \), so we can rewrite \( y \) as:

\[
y = 3u
\]

Now we need to differentiate \( y \) with respect to \( x \). We'll use the chain rule, which states:

\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]

1. **Differentiate \( y \) with respect to \( u \):**
\[
\frac{dy}{du} = 3
\]

2. **Differentiate \( u \) with respect to \( x \):**
To find \( \frac{du}{dx} \), we need to differentiate \( u = 2^{2x} \). Using the chain rule again:
- The derivative of \( 2^{v} \) with respect to \( v \), where \( v = 2x \), is \( 2^{v} \ln(2) \).
- We also need to differentiate \( 2x \) with respect to \( x \), which is \( 2 \).

Therefore, the derivative of \( u = 2^{2x} \) is:
\[
\frac{du}{dx} = 2^{2x} \cdot \ln(2) \cdot 2 = 2^{2x} \cdot 2 \ln(2)
\]

3. **Combine using the chain rule:**
Now, we substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3 \cdot (2^{2x} \cdot 2 \ln(2))
\]

Simplifying that gives:
\[
\frac{dy}{dx} = 6 \ln(2) \cdot 2^{2x}
\]

Thus, the derivative of the function \( y = 3(2^{2x}) \) is:

\[
\frac{dy}{dx} = 6 \ln(2) \cdot 2^{2x}
\]