# QUESTIONS ON DIFFERENTIATION FOR CLASS 11

## About "Questions on Differentiation for Class 11"

Questions on Differentiation for Class 11 :

Here we are going to see some practice questions on differentiation.

Find the derivatives of the following functions with respect to corresponding independent variables:

Question 1 :

Differentiate y = (tan x -  1)/ sec x

Solution :

y = (tan x -  1)/ sec x

u = tan x - 1  ===> u'  =  sec2 x - 0

v = sec x ===> v'  =  sec x tan x

Quotient rule :

d(u/v)  =  (vu' - uv') / v2

=  [sec x(sec2 x) - (tan x - 1)sec x tan x]  / sec2 x

=  [sec3 x - tan2x sec x + sec x tan x]  / sec2 x

=  [sec x(sec2x - tan2x) + sec x tan x]  / sec2 x

dy/dx  =  [sec x(1) + sec x tan x]  / sec2 x

dy/dx  =  sec x(1 + tan x)/ sec2x

dy/dx  =  (1 + (sin x/cos x))/(1/cos x)

dy/dx  =  ((cos x + sin x)/cos x) ⋅ (cos x / 1)

dy/dx  =  cos x + sin x

Question 2 :

Differentiate y = sin x / x2

Solution :

y = sin x / x2

u = sin x  ==> u'  =  cos x

u = x2  ==> v'  =  2 x

dy/dx  =  [x2 (cos x) - sin x(2x)]/(x2)2

dy/dx  =  [x2 (cos x) - 2x sin x]/x4

dy/dx  =  x [x cos x - 2 sin x]/x4

dy/dx  =  [x cos x - 2 sin x]/x3

Question 3 :

Differentiate y = tan θ (sin θ + cos θ)

Solution :

y = tan θ (sin θ + cos θ)

u = tan θ  ==> u'  =  sec2 θ

v = sin θ + cos θ ==> v'  =  cos θ - sin θ

dy/dx  =  tan θ (cos θ - sin θ) + (sin θ + cos θ) (sec2θ)

=  tan θ cos θ - tan θ sin θ + sin θ sec2θ + cos θ sec2θ

=  sin θ - (sin2 θ/cos θ) + (sin θ/cos2θ) + (1/cos θ)

=  sin θ - (sin2 θ/cos θ) + (tan θ sec θ) + (1/cos θ)

=  sin θ - ((1 - sin2 θ)/cos θ) + (tan θ sec θ)

=  sin θ - (cos2 θ/cos θ) + (tan θ sec θ)

dy/dx  =  sin θ - cos θ + tan θ sec θ

Question 4 :

Differentiate y = cosec x ⋅ cot x

Solution :

y = cosec x ⋅ cot x

u = cosec x  ==> u'  =  -cosec x cot x

v = cot x ==> v'  =  -cosec2 x

dy/dx  =  cosec x (-cosec2 x) + cot x(-cosec x cot x)

=  - cosec3 x - cosec x cot2

=  - (1/sin3 x) - (1/sin x) (cos2 x/sin2 x)

=  - (1/sin3 x) - (cos2 x/sin3 x)

dy/dx   =  - (1 + cos2x) /sin3 x

After having gone through the stuff given above, we hope that the students would have understood, "Questions on Differentiation for Class 11"

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