Question

(a) Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x>0$.

Hint:

When differentiating functions involving nested exponents, apply the chain rule carefully for each layer.

Answer 1

We are asked to differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$. Let: \[ y = \sqrt{e^{\sqrt{2x}}}. \] This simplifies to: \[ y = e^{\frac{\sqrt{2x}}{2}}. \] Now, differentiate $y$ with respect to $e^{\sqrt{2x}}$. Since we are differentiating with respect to $e^{\sqrt{2x}}$, we use the chain rule: \[ \frac{dy}{de^{\sqrt{2x}}} = \frac{1}{2} e^{\frac{\sqrt{2x}}{2}}. \] This is the required differentiation.