Vinod Singh and Shweta Pawar Solutions for Chapter: Differentiation, Exercise 2: Critical Thinking

If $x=a\left(1-\cos \theta \right)$ and $y=a\left(\theta +\sin \theta \right)$, then $\frac{d^{2}y}{d{x}^2}$ at $\theta =\frac{\pi }{2}$ is

The $2^{nd}$ derivative of  $a$$\sin {}^{3}t$ with respect to $a{\cos }^{3}t$ at $t=\frac{\pi }{4}$ is

If $e^y+xy=e$, then the value of $\frac{d^{2}y}{d{x}^2}$ for $x=0$, is

The derivative of every even function

If $f\left(x\right)$ is an odd differentiable function defined on $\left(-\infty ,\infty \right)$ such that $f^{\text{'}}\left(3\right)=2$, then $\left|f^{\text{'}}\left(-3\right)\right|$ is

If $\left(\frac{y}{x}\right)+\left(\frac{x}{y}\right)=2$, then $\frac{dy}{dx}$ is equal to

If $y=e^x\cdot e^{2x}\cdot e^{3x}\mathrm{....}{e}^{nx}$, then $\frac{dy}{dx}=$

If $y=\sqrt{\frac{1+e^x}{1-e^x}}$, then  $\frac{dy}{dx}=$