QUESTIONS ON DIFFERENTIATION FOR CLASS 11

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 x 

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

  =  - (1/sin³ x) - (cos² x/sin³ x)

dy/dx   =  - (1 + cos²x) /sin³ x

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