Amit M Agarwal Solutions for Chapter: Differentiation, Exercise 4: Target Exercises

The functions $u=e^x\sin x, v=e^x\cos x$ satisfy the equation

The derivative of ${cosec}^{-1}\left(\frac{1}{2x^2-1}\right)$ w.r.t. $\sqrt{1-x^2}$ at $x=\frac{1}{2}$ is equal to

The derivative of ${\sin }^{-1}{\left(3x-4x\right)}^3$ w.r.t. ${\sin }^{-1}x$ is equal to

If a curve is given by $x=a\cos t+\frac{b}{2}\cos 2t$ and $y=a\sin t+\frac{b}{2}\sin 2t,$ then the points for which $\frac{d^{2}y}{d{x}^2}=0,$ are given by

Let $f$ be a differentiable function satisfying $f\left(x\right)+f\left(y\right)+f\left(z\right)+f\left(x\right)f\left(y\right)f\left(z\right)=14,\forall$ $x, y, z\in R.$ Then,

Let $f$ be a real function such that $f\left(x-y\right), f\left(x\right) f\left(y\right)$ and $f\left(x+y\right)$ are in $\mathrm{AP}$ for all $x, y\in R.$ If $f\left(0\right)\ne 0,$ then

If $f\left(x\right)$ and $g\left(x\right)$ are two differentiable functions on $\left[0,2\right]$ such that $f^{"}\left(x\right)-g^{"}\left(x\right)=0,$ $f^{\text{'}}\left(1\right)=4, g^{\text{'}}\left(1\right)=2$ and $f\left(2\right)=9, g\left(2\right)=3$, then $f\left(x\right)-g\left(x\right)$ at $x=\frac{3}{2}$ is equal to

Let $h\left(x\right)=g^{\text{'}}\left(x\right)$, where $g\left(x\right)$ is the inverse function of $f\left(x\right)$ and $f\left(x\right)$ is invertible and $f^{\text{'}}\left(x\right)=\frac{f^2\left(x\right)-1}{f\left(x\right)+\left(f^2\left(x\right)-1\right)e^{af\left(x\right)}\sin \left(2f\left(x\right)+3\right)},$ then $h^{\text{'}}\left(x\right)$ is equal to