Question

Differentiation question help. *question attached below* Will give medal

Answer 1

Can I have some help with an explanation as to how to do this question, please?

Answer 2

\[
\text{Find }\frac{d}{dx}x^{\sqrt{x}} \\
\text{Let } y = x^{\sqrt{x}}. ~~\text{Find } \frac{dy}{dx} \\
\text{Take logarithm on both sides:} \\
\ln(y) = \sqrt{x}\ln(x) \\
\text{Differentiate both sides: } \\
\frac 1yy' = \sqrt{x}\frac 1x + \ln(x)\frac 12 x^{-1/2} =
\frac{\sqrt{x}}{x} + \frac 12\frac{\ln(x)}{\sqrt{x}} =
\frac{1}{\sqrt{x}} + \frac 12\frac{\ln(x)}{\sqrt{x}} = \\
\frac{1}{\sqrt{x}}*\frac 22 + \frac 12\frac{\ln(x)}{\sqrt{x}} =
\frac{2+\ln(x)}{2\sqrt{x}} \\
\frac 1yy' = \frac{2+\ln(x)}{2\sqrt{x}} \\
y' = y \left[ \frac{2+\ln(x)}{2\sqrt{x}} \right ] \\
\frac{dy}{dx} = x^{\sqrt{x}} \left[ \frac{2+\ln(x)}{2\sqrt{x}} \right ]
\]

Answer 3

Wow, thank you so much for the steps and explanation!