Questions: y=2^(4x^4)

Transcript text: $y=2^{4 x^{4}}$

Solution

Solution Steps

Step 1: Differentiate the Function

We start with the function $ y = 2^{4x^4} $. To find the derivative $ \frac{dy}{dx} $, we apply the chain rule. The derivative of $ a^u $ is given by $ a^u \ln a \cdot \frac{du}{dx} $. Here, $ u = 4x^4 $, so we first compute $ \frac{du}{dx} $.

Step 2: Compute $ \frac{du}{dx} $

Calculating the derivative of $ u $: \[ \frac{du}{dx} = \frac{d}{dx}(4x^4) = 16x^3 \]

Step 3: Apply the Chain Rule

Now we can apply the chain rule: \[ \frac{dy}{dx} = 2^{4x^4} \ln(2) \cdot \frac{du}{dx} = 2^{4x^4} \ln(2) \cdot 16x^3 \] This simplifies to: \[ \frac{dy}{dx} = 16x^3 \cdot 2^{4x^4} \ln(2) \]

Final Answer

The derivative of the function $ y = 2^{4x^4} $ is: \[ \boxed{\frac{dy}{dx} = 16x^3 \cdot 2^{4x^4} \ln(2)} \]

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