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

Solution Steps

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 $\frac{du}{dx}$.

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

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