R S Aggarwal Solutions for Exercise 8: EXERCISE

The differentiation of $e^{\sin x}$  with respect to $\cos x$ is of the form $-a{e}^{\sin x}·\cot x$ then find the value of $a.$

If differentiation of ${\tan }^{-1}\sqrt{\frac{1-x^2}{1+x^2}}$ with respect to ${\cos }^{-1}{x}^2$ is of the form $\frac{1}{k}$ then find the value of $k$.

Differentiation of ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$ with respect to ${\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$ is 

If on differentiating ${\tan }^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ with respect to ${\cos }^{-1}\left(2x^2-1\right)$ we get the answer in $-\frac{a}{b}$ form then write the value of $a+b$.

Differentiate ${\sin }^{3}x$ with respect to ${\cos }^{3}x$

If on differentiating ${\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ with respect to ${\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)$ we get the value of the form $\frac{m}{n}$ then find $m+n$.

If differentiation of ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to ${\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$ is $\frac{a}{b}$ then find the value of $a+b$.

If on differentiating ${\tan }^{-1}\left(\frac{\sqrt{1-{\mathrm{x}}^2}}{\mathrm{x}}\right)$ with respect to ${\cos }^{-1}\left(2x\sqrt{1-x^2}\right)$ when $\mathrm{x}\ne 0$ we get the value of the form $\frac{a}{b}$ then find $a+b$.