First Principle of Differentiation

Let$f\left(x\right)=x+\frac{1}{2x+\frac{1}{2x+\frac{1}{2x+\dots \dots \dots \infty }}}$ Then the value of$f\left(100\right)·f^{\text{'}}\left(100\right)$ is 

Find by first principle, the derivative of function $\cos \left(\log x\right)$.

Find the derivative of the function $f\left(x\right)=\cos 2x$, using the method of first principle.

Find the derivative of the function $f\left(x\right)=\sin 3x$, using the method of first principle.

Find the derivative of the function $f\left(x\right)=100x^{100}$.

Find the derivative of $\tan \left(3x+7\right)$.

On differentiating $\mathrm{x} \cos \mathrm{x}$ w.r.t $\mathrm{x}$ using first principle, we get $-\mathrm{xsinx}\odot \mathrm{cosx}$. What should be the mathemetical operator that should replace $\odot$

Differentiate ${\mathrm{x}}^{2/3}$ w.r.t $\mathrm{x}$ using first principle we get $\frac{k}{3}{\mathrm{x}}^{-1/3}$ then find the value of $k$.

Derivative of $\frac{\mathrm{ax}+\mathrm{b}}{\mathrm{cx}+\mathrm{a}}$ is $\frac{ad-k}{(cx+d{)}^2}$. Find $k$.

If Differentiation of $\cos \left({\mathrm{x}}^2+1\right)$ w.r.t $\mathrm{x}$ using the first principle is $k\sin \left({\mathrm{x}}^2+1\right),$ then the value of $k$ is:

Differentiate of  $\cos \left({\mathrm{x}}^2+1\right)$ is  $-K\mathrm{xsin}\left({\mathrm{x}}^2+1\right)$ then find the value of $K$

Let $y=\sec x+\tan x,$ then $\frac{dy}{dx}$ at the point $x=\frac{\pi }{3},$ is:

A differentiable function $\text{'}f\text{'}$ is satisfying the relation $f(x+y)=f\left(x\right)+f\left(y\right)+2xy(x+y)-\frac{1}{3}\forall x,y\in R$ and $\underset{h\to 0}{\lim }\frac{3f\left(h\right)-1}{6h}=\frac{2}{3}.$ Then

If $f\left(x\right)=|x–2|$ and $g\left(x\right)=f\left[f\left(x\right)\right],$ then $g\text{'}\left(x\right)=...$ for $x>20$.

Let $f\left(x+y\right)=f\left(x\right)f\left(y\right)$ for all $x$ and $y.$ If $f\left(0\right)=1,f\left(3\right)=3$ and $f^{\prime }\left(0\right)=11,$ then the value of $f^{\text{'}}\left(3\right)$ is ______

If $\left|f\left(x\right)-f\left(y\right)\right|\le 2{\left|x-y\right|}^{\frac{3}{2}} \forall x,y\in R$ and $f\left(0\right)=1$, then value of $\underset{0}{\overset{1}{\int }}{f}^2\left(x\right)dx$ is equal to

If $f\left(x\right)=\left|\cos x-\sin x\right|,$ then $f^{\prime }\left(\frac{\pi }{6}\right)$ is equal to

$\left(\frac{d}{dx}\right)\left[\log \left(\sec x-\tan x\right)\right]$ is equal to