Vinod Singh and Shweta Pawar Solutions for Chapter: Differentiation, Exercise 3: Competitive Thinking

Let $g\left(x\right)$ be the inverse of the function $f\left(x\right)$ and $f^{\text{'}}\left(x\right)=\frac{1}{1+x^3},$ then $g^{\text{'}}\left(x\right)$ is equal to

Let $f\left(x\right)={\tan }^{-1}x$. Then, $f^{\text{'}}\left(x\right)+f^{\text{'}\text{'}}\left(x\right)$ is equal to $0,$ when $x$ is equal to

If $x=a\left(t-\frac{1}{t}\right),y=a\left(t+\frac{1}{t}\right),$ then $\frac{dy}{dx}=$

If $2y={\sin }^{-1}\left(x+5y\right)$, then $\frac{dx}{dy}$ is equal to

Let $f:R\to R$ be a function, such that $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$ for all $x, y\in R$. If $f\left(x\right)$ is differentiable at $x=0,$ then which one of the following is incorrect?

If $x^2+y^2=t+\frac{2}{t}$ and $x^4+y^4=t^2+\frac{4}{t^2},$ then  $x^{3}y\frac{dy}{dx}=$

Let $f_1\left(x\right)=e^x, f_2\left(x\right)=e^{f_1\left(x\right)}, \dots , f_{n+1}\left(x\right)=e^{f_n\left(x\right)}$ for all $n\ge 1$. Then, for any fixed $n,\frac{d}{dx}{f}_n\left(x\right)$ is

Let $f:R\to R$ be a differentiable function. If $f$ is even, then $f^{\text{'}}\left(0\right)$ is equal to