Class 12 NCERT Solutions- Mathematics Part I – Chapter 5 Continuity And Differentiability – Exercise 5.2

 Differentiate the function with respect to x in Question 1 to 8

Question 1. Sin(x² + 5)

Solution: 

y = sin(x² + 5)

 = 

= cos(x² + 5) × 

= cos(x² + 5) × (2x)

dy/dx = 2xcos(x² + 5)

Question 2. cos(sin x)

Solution:

y = cos(sin x)

 = 

 = -sin(sin x) × 

= -sin(sin x)cos x  

Question 3. sin(ax + b)

Solution:

y = sin(ax + b)

= a cos(ax + b)  

Question 4. Sec(tan(√x)

Solution:

y = sec(tan√x)

 = 

= sec(tan √x) × tan(√x) × 

= sec (tan √x) × tan (tan √x) × sec2√x × 

= sec(tan√x)tan(tan√x)(sec2√x)1/(2√x)

= 1/(2√x) × sec(tan√x)tan(tan√x)(sec2√x)

Question 5. 

Solution:

y = 

=

Question 6. cos x³.sin²(x⁵)

Solution:

y = cos x³.sin²(x⁵)

= 

= cos x³.2sin(x⁵) .cos(x⁵(5x⁴)(5x⁴) – sin²(x⁵).sin x³.3x²

= 10x⁴ cos x³sin(x⁵)cos(x⁵) – 3x² sin²(x⁵)sin x³

Question 7. 2√(cos(x²))

Solution:

y = 2√(cos(x²))

 = 

= 2

= 

= 

= 

=

= 

= 

= 

= 

= 

= 

= 

Question 8. cos (√x)

Solution:

y = cos (√x)

dy/dx = -sin√x

=

=

Question 9. Prove that the function f given by f(x) = |x – 1|, x ∈ R is not differentiable at x = 1.

Solution:

= 

= 

=

= +1                                                                                                            

= 

= 

= 

= -1   

LHD ≠ RHD  

Hence, f(x) is not differentiable at x = 1  

Question 10. Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.

Solution:

Given: f(x) = [x], 0 < x < 3

LHS:

f'(1) = 

= 

=

= ∞

RHS:

f'(1) = 

= 

= 

= 

= 0

LHS ≠ RHS

So, the given f(x) = [x] is not differentiable at x = 1. 

Similarly, the given f(x) = [x] is not differentiable at x = 2.