If $e^{x}=y+\sqrt{y^2-1}$,then $\frac{d y}{d x}=$

The value of $\frac{d}{d x}\left[\log \left(\sin \sqrt{\frac{x^2+1}{x^2+2}}\right)\right]$ when $x=\sqrt{2}$ is:

If $y = \cot^{-1}(x^2)$,then $\frac{dy}{dx}$ is equal to

Consider the function $f: [1.2, 1.9] \rightarrow \mathbb{R}$ defined by $f(x) = [x]$,where $[x]$ denotes the greatest integer less than or equal to $x$. Which of the following is true?

Find the derivative of the function $f(x)=2x^{2}+3x-5$ at $x=-1$. Also,prove that $f^{\prime}(0)+3f^{\prime}(-1)=0$.

The differential of $f(x) = \log_{e}(1 + e^{10x}) - \tan^{-1}(e^{5x})$ at $x = 0$ and for $dx = 0.2$ is