Embibe Experts Solutions for Chapter: Differential Coefficient, Exercise 3: EXERCISE-3

If $y={\tan }^{-1}\frac{u}{\sqrt{1-u^2}}$ and $x={\sec }^{-1}\frac{1}{2u^2-1},u\in \left(0,\frac{1}{\sqrt{2}}\right)\cup \left(\frac{1}{\sqrt{2}},1\right)$; find the value of $\left|\frac{dy}{dx}\right|$.

If $y={\tan }^{-1}\frac{x}{1+\sqrt{1-x^2}}+\sin \left(2{\tan }^{-1}\sqrt{\frac{1-x}{1+x}}\right)$ for $x\in \left(-1,1\right)$ and if the value of $\frac{dy}{dx}$ at $x=0$ is $k$, then find the value of $2k$

If $x=2 \cos t-\cos 2t$ and $y=2 \sin t-\sin 2t$, find the value of $\left|\frac{d^{2}y}{d{x}^2}\right|$ when $t=\left(\frac{\pi }{2}\right)$.

Let $y=\frac{a{x}^2}{\left(x-a\right)\left(x-b\right)\left(x-c\right)}+\frac{bx}{\left(x-b\right)\left(x-c\right)}+\frac{c}{\left(x-c\right)}+1$, if $\frac{y^{\text{'}}}{y}=\frac{\lambda }{x}\left(\frac{a}{a-x}+\frac{b}{b-x}+\frac{c}{c-x}\right)$, then find the value of $\lambda$.

Let $f\left(x\right)=x^2-4x-3, x>2$ and let $g$ be the inverse of $f$. Find the value of $g^{\text{'}}$ at $f\left(x\right)=9$ .

Let $y=x\ln \left[{\left(ax\right)}^{-1}+a^{-1}\right]$, if $x\left(x+1\right)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=y-\mu$, then find the value of $\mu$.

Suppose $f$ and $g$ are two functions such that $f, g:R\to R, f\left(x\right)=\ln \left(1+\sqrt{1+x^2}\right)$ and $g\left(x\right)=\ln \left(x+\sqrt{1+x^2}\right)$, then find the value of $x.e^{g\left(x\right)}{\left(f\left(\frac{1}{x}\right)\right)}^{\text{'}}+g^{\text{'}}\left(x\right)$ at $x=1$.

If $f\left(x\right)=\left|\begin{array}{ccc}{\left(x-a\right)}^4& {\left(x-a\right)}^3& 1\\ {\left(x-b\right)}^4& {\left(x-b\right)}^3& 1\\ {\left(x-c\right)}^4& {\left(x-c\right)}^3& 1\end{array}\right|$ then $f^{\text{'}}\left(x\right)=\lambda .\left|\begin{array}{ccc}{\left(x-a\right)}^4& {\left(x-a\right)}^2& 1\\ {\left(x-b\right)}^4& {\left(x-b\right)}^2& 1\\ {\left(x-c\right)}^4& {\left(x-c\right)}^2& 1\end{array}\right|$. Find the value of $\lambda$ .