Mathematics - Application of Derivative Question with Solution | TestHub

$f(x)=\frac{x}{\sqrt{a^2+x^2}}-\frac{(d-x)}{\sqrt{b^2+(d-x{)}^2}}$ $=\frac{x}{\sqrt{a^2+x^2}}+\frac{(x-d)}{\sqrt{b^2+(x-d{)}^2}}$ $\begin{array}{l}{f}^{\prime }(x)=\frac{\sqrt{a^2+x^2}-\frac{x(2x)}{2\sqrt{a^2+x^2}}}{\left(a^2+x^2\right)}\\ =\frac{a^2+x^2-x^2}{{\left(a^2+x^2\right)}^{3/2}}+\frac{b^2+(x-d{)}^2-(x-d{)}^2}{{\left(b^2+(x-d{)}^2\right)}^{3/2}}\\ +\frac{\sqrt{b^2+(x-d{)}^2}-\frac{(x-d)2(x-d)}{2\sqrt{b^2+(x-d{)}^2}}}{\left(a^2+(x-d{)}^2\right)}\\ \Rightarrow f^{\prime }(x)>0,\square x\in R\end{array}$ $\Rightarrow f(x)$ is increasing function. Hence, $f(x)$ is increasing function