Chapter 5 Continuity and Differentiability Ex 5.2 Solutions
Question - 1 : - Differentiatethe functions with respect to x.-
Answer - 1 : - Let y = sin(x2 + 5),
put x² + 5 = t
y = sint
t = x²+5
_= cos (x² + 5) × 2x
_= 2x cos (x² + 5)Differentiatethe functions with respect to x.
Answer - 2 : -
Thus, f isa composite function of two functions.
Put t = u (x)= sin x
By chain rule,
Alternate method
Differentiatethe functions with respect to x.
Answer - 3 : - 
Thus, f isa composite function of two functions, u and v.
Put t = u (x) = ax + b
Hence, by chain rule, we obtain
Alternate method
Differentiatethe functions with respect to x.
Answer - 4 : - 
Thus, f isa composite function of three functions, u, v, and w.
Hence, by chain rule, we obtain
Alternate method
Differentiatethe functions with respect to x.
Answer - 5 : - The given function is
,where g (x) = sin (ax + b) and
h (x) =cos (cx + d)
∴ g is a composite functionof two functions, u and v.
Therefore, by chain rule, we obtain
∴h is a composite function of twofunctions, p and q.
Put y = p (x)= cx + d
Therefore, by chain rule, we obtain
Differentiatethe functions with respect to x.
Answer - 6 : - The given function is
Question - 7 : - Differentiatethe functions with respect to x.
Answer - 7 : -
Question - 8 : - Differentiatethe functions with respect to x.
Answer - 8 : - 
Clearly, f isa composite function of two functions, u and v,such that
By using chain rule, we obtain
Alternate method
Provethat the function f given by
isnotdifferentiable at x = 1.
Answer - 9 : - The given function is
It isknown that a function f is differentiable at a point x = c inits domain if both
are finite and equal.
Tocheck the differentiability of the given function at x = 1,
considerthe left hand limit of f at x = 1
Sincethe left and right hand limits of f at x = 1are not equal, f is not differentiable at x =1
is notdifferentiableat x = 1 and x = 2.
Answer - 10 : - The given function f is
It isknown that a function f is differentiable at a point x = c inits domain if both.
are finite and equal.
Tocheck the differentiability of the given function at x = 1,consider the left hand limit of f at x = 1
Sincethe left and right hand limits of f at x = 1are not equal, f is not differentiable at
x = 1
Tocheck the differentiability of the given function at x = 2,consider the left hand limit
of f at x =2
Sincethe left and right hand limits of f at x = 2are not equal, f is not differentiable at x =2