Question

Use logarithmic differentiation

sqrt(x+1)/cubedroot(x-1)

for the answer I get 1/2sqrt(x+1)-1/3(x-1)^2/3

but my book says its wrong that it is also multiplied by orgional equation

Answer 1

\[\begin{align*}
y&=\frac{\sqrt{x+1}}{\sqrt[3]{x-1}}\\\\
\ln y&=\ln\bigg(\cdots\bigg)\\\\
&=\frac{1}{2}\ln(x+1)-\frac{1}{3}\ln(x-1)\\\\
\frac{1}{y}\frac{dy}{dx}&=\frac{1}{2}\frac{1}{x+1}-\frac{1}{3}\frac{1}{x-1}\\\\
\frac{dy}{dx}&=y\left[\frac{1}{2}\frac{1}{x+1}-\frac{1}{3}\frac{1}{x-1}\right]\\\\
&=\frac{\sqrt{x+1}}{\sqrt[3]{x-1}}\left[\frac{1}{2}\frac{1}{x+1}-\frac{1}{3}\frac{1}{x-1}\right]\end{align*}\]