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

The objective is to differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$. Let \( y = \sqrt{e^{\sqrt{2x}}} \). This expression can be simplified to \( y = e^{\frac{\sqrt{2x}}{2}} \). Applying the chain rule for differentiation with respect to $e^{\sqrt{2x}}$, we get: \[ \frac{dy}{de^{\sqrt{2x}}} = \frac{1}{2} e^{\frac{\sqrt{2x}}{2}}. \] This result represents the required differentiation.