Embibe Experts Solutions for Chapter: Differential Coefficient, Exercise 4: EXERCISE-4

If $\left|x\right|<1$, then prove that $\frac{1-2x}{1-x+x^2}+\frac{2x-4x^3}{1-x^2+x^4}+\frac{4x^3-8x^7}{1-x^4+x^8}+\dots \dots \infty =\frac{1+2x}{1+x+x^2}$.

Differentiate $y={\cos }^{-1}\frac{1-x^2}{1+x^2}$ w.r.t. ${\tan }^{-1}x,$ stating clearly where function is not differentiable.

If $x=\frac{1}{z}$ and $y=f\left(x\right)$, show that: $\frac{d^{2}f}{d{x}^2}=2z^3\frac{dy}{dz}+z^4\frac{d^{2}y}{d{z}^2}$

Prove that if $\left|a_1\sin x+a_2\sin 2x+....+a_n\sin nx\right|\le \left|\sin x\right|$ for $x\in R$, then $\left|a_1+2a_2+3a_3+....+n{a}_n\right|\le 1$

Show that the substitution $z=\ln \left(\tan \frac{x}{2}\right)$ changes the equation $\frac{d^{2}y}{d{x}^2}+\cot x\frac{dy}{dx}+4y {\mathrm{cosec}}^{2}x=0$  to  $\left(\frac{d^{2}y}{d{z}^2}\right)+4y=0$.

Find a polynomial function $f\left(x\right)$ such that $f\left(2x\right)=f^{\text{'}}\left(x\right)f^{"}\left(x\right)$.

If $Y=sX$ and $Z=tX$, where all the letters denotes the function of $x$ and suffixes denotes the differentiation w.r.t. $x$ then prove that $\left|\begin{array}{ccc}X& Y& Z\\ X_1& Y_1& Z_1\\ X_2& Y_2& Z_2\end{array}\right|=X^3\left|\begin{array}{cc}{s}_1& t_1\\ s_2& t_2\end{array}\right|$

If $y={\tan }^{-1}\frac{1}{x^2+x+1}+{\tan }^{-1}\frac{1}{x^2+3x+3}+{\tan }^{-1}\frac{1}{x^2+5x+7}+{\tan }^{-1}\frac{1}{x^2+7x+13}+....$ upto $n$ terms. Find $\frac{dy}{dx}$, expressing your answer in $2$ terms.