Differentiability

If function is differentiable at $x = a$, find $\underset{x\to a}{\lim }\frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$.

If $f\left(x\right)=\left|x-2\right|+\left|x-5\right|$ then find $f\text{'}\left(4\right)$.

If $F(x,y,z)=0$, prove $\frac{dz}{dy}.\frac{dy}{dx}.\frac{dx}{dz}=-1$

If the function 

$f\left(x\right)=\left\{\begin{array}{l}-x x<1\\ a+{\cos }^{-1}(x+b) 1\le x\le 2\end{array}\right.$is differentiable at x=1 then $\frac{a}{b}=$

If $f\left(x\right)=\left\{\begin{array}{ll}3,& x<0\\ 2x+1,& x⩾0\end{array}\right.$, then

If $f\left(x+y\right)=f\left(x\right)f\left(y\right)$ for all $x$ and $y$ and $f\left(x\right)=1+g\left(x\right)G\left(x\right)$ where $\underset{x\to 0}{\lim } g\left(x\right)=0$ and $\underset{x\to 0}{\lim } G\left(x\right)=0$ exists. Prove that $f\left(x\right)$ is continuous for all $x.$

If the Left Hand Derivative and Right Hand Derivative of a function is not equal but finite then the function must be a continuous function.

If $f\left(x\right)=\left\{\begin{array}{l}x+\sin x+a x\ge 0\\ x^2+bx+b x<0\end{array}\right.$ is differentiable at $x=0$ then $a+b$ equals

Find the number of points where the function $f\left(x\right)=Max.\left\{\left|\tan x\right|,\cos \left|x\right|\right\}$ where function is not differentiable in $\left[-\mathrm{\pi },\mathrm{\pi }\right]$.

Find the set of points where $f\left(x\right)=x\left|x\right|$ is differentiable

Let $f\left(x\right)=\left\{\begin{array}{ll}{x}^2\sin \frac{1}{x},& x\ne 0\\ 0,& x=0\end{array}\right.$. Then 

Find the slope of the tangent at a point $\left(7,78\right)$ to the curve $y=x^2+4x+1$.

Find the slope of the tangent at a point $\left(-5,-6\right)$ to the curve $y=x^5+2x^2+1$.

Find the slope of the tangent at a point $\left(1,2\right)$ to the curve $y=4x^3$.

Find the slope of the tangent at a point $\left(5,6\right)$ to the curve $y=x^4+2x^2+x$.

Find the slope of the tangent at a point $\left(3,4\right)$ to the curve $y=x^3+3x^2+1$.

The value of the right-hand derivative of $\mathrm{f}\left(\mathrm{x}\right)$ at $x=2$ is ; $f\left(x\right)=$$\left\{\begin{array}{ll}1+x, \mathrm{when}& x\le 2\\ 5–x, \mathrm{when}& x>2\end{array}\right.$

Find the right hand derivative and the left hand derivative of $$\begin{gathered} f\left(x\right)=x-1, x<2 \\ =2x-3, x\ge 2 \end{gathered}$$ at the point $x=2$ and hence show that $f\left(x\right)$ is not differentiable at $x=2$.

Given $f\left(x\right)=\left\{\begin{array}{l}{x}^2{e}^{2(x-1)},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0}\le x\le 1.\\ \cos (2x-2)+b{x}^2,1<x\le 2\end{array}\right.$

If $f\left(x\right)$ is differentiable at $x=1$ then the value of $\left|a-b\right|$ is

Discuss the differentiability of the function

$f\left(x\right)=\left\{\begin{array}{l}2x-1 \mathrm{when} x<\frac{1}{2}\\ 3-6x \mathrm{when} x\ge \frac{1}{2}\end{array}\right.$  at $x=\frac{1}{2}$.