# 5 - Differentiation Questions Answers

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

then a=?, b=?

**Asked By: RAJIV**

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**Joshi sir comment**

1+x^{2}+x^{4} = (1+x+x^{2})(1-x+x^{2})

now solve

**Asked By: GAURAV MAHATE**

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**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 ∑ ^{n}_{r=1 }1/n e^{r/n }n tends to infinity.

**Asked By: NIKHIL VARSHNEY**

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**Joshi sir comment**

limn->∞ ∑_{r=1}^{n }e^{r/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

_{0}∫^{1} e^{x}dx

now solve it

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

**Asked By: NIKHIL VARSHNEY**

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**Joshi sir comment**

f(x) = x|x| => f(x) = -x^{2} for negative real values of x and f(x) = x^{2} 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

**Asked By: NIKHIL VARSHNEY**

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**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 = (1-2sin^{2}h+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