# 5 - Differentiation Questions Answers

d/dx((1+x^2+x^4)/(1+x+x^2))=ax+b

then a=?, b=?

Joshi sir comment

1+x2+x4 = (1+x+x2)(1-x+x2)

now solve

Sir,is the function (2x+5) is differentiable everywhere in its domain set. If yes the what is the case with l2x+5l.
Joshi sir comment

since first one is a straight line so at every point only one tangent is possible so it is differentiable

but second one which is modulus has a sharp turn at x = -5/2 so two tangents at x = -5/2 are possible so is not differentiable.

lim ∑ nr=1 1/n er/n n tends to infinity.

Joshi sir comment

limn->∞ r=1er/n/n

This question is related to definite integration, consider 1 and divide it into n parts, upto nth part total value = r/n is equivalent to x and 1/n is dx

so question will be

01 exdx

now solve it

if f(X)=xlxl then find f-1(x) can you please explain me the meaning and use of sgn

Joshi sir comment

f(x) = x|x|  => f(x) = -x2  for negative real values of x  and f(x) = x2 for positive real values of x

so f-1(x) = -√|x|  for negative real values and = +√|x| for positive real values of x

which of the following is differntiable at x=0 ?

cos(lxl)+lxl

sin(lxl)-lxl

Joshi sir comment

sin(|x|)+|x| is differentiable at x = 0

LHD of first = {cos|0+h|+|0+h|-cos|0|-|0|}/0+h-0  = (cosh+h-1)/h = (1-2sin2h+h-1)/h = 1 (on taking limits)

now check RHD, its value will be -1

similarly in second both values are 0 so it is differentiable