These NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 Questions and Answers are prepared by our highly skilled subject experts.
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.2
Question 1.
sin(x² + 5)
Solution:
Let y = sin(x² + 5),
put x² + 5 = t
y = sint
t = x²+5
\(\frac { dy }{ dx } =\frac { dy }{ dt } .\frac { dt }{ dx } \)
\(\frac { dy }{ dx } =cost.\frac { dt }{ dx } =cos({ x }^{ 2 }+5)\frac { d }{ dx } ({ x }^{ 2 }+5)\)
= cos (x² + 5) × 2x
= 2x cos (x² + 5)
Question 2.
cos (sin x)
Solution:
let y = cos (sin x)
put sinx = t
∴ y = cost,
t = sinx
∴\(\frac { dy }{ dx } =-sin\quad t,\frac { dt }{ dx } =cos\quad x \)
\(\frac { dy }{ dx } =\frac { dy }{ dt } .\frac { dt }{ dx } =(-sint)\times cosx\)
Putting the value of t, \(\frac { dy }{ dx } =-sin(sinx)\times cosx\)
\(\frac { dy }{ dx } =-[sin(sinx)]cosx\)
Question 3.
sin(ax+b)
Solution:
let = sin(ax+b)
put ax+bx = t
∴ y = sint
t = ax+b
\(\frac { dy }{ dt } =cost,\frac { dt }{ dx } =\frac { d }{ dx } (ax+b)=a\)
\(Now\frac { dy }{ dx } =\frac { dy }{ dt } .\frac { dt }{ dx } =cost\times a=acos\quad t\)
\(\frac { dy }{ dx } =acos(ax+b)\)
Question 4.
sec(tan(\(\sqrt{x}\)))
Solution:
let y = sec(tan(\(\sqrt{x}\)))
by chain rule
\(\frac { dy }{ dx } =sec(tan\sqrt { x } )tan(tan\sqrt { x } )\frac { d }{ dx } (tan\sqrt { x } )\)
\(\frac { dy }{ dx } =sec(tan\sqrt { x } ).tan(tan\sqrt { x } ){ sec }^{ 2 }\sqrt { x } .\frac { 1 }{ 2\sqrt { x } } \)
Question 5.
\(\\ \frac { sin(ax+b) }{ cos(cx+d) } \)
Solution:
Question 6.
cos x³ . sin²(x5) = y(say)
Solution:
Let y = cos x³ . sin²(x5)
Question 7.
\(2\sqrt { cot({ x }^{ 2 }) }\)
Solution:
Question 8.
cos(\(\sqrt{x}\))
Solution:
Question 9.
Prove that the function f given by f (x) = |x – 1|, x ∈ R is not differential at x = 1.
Solution:
∴ f is not differentiable at x = 1
Question 10.
Prove that the greatest integer function defined by f (x) = [x], 0 < x < 3 is not differential at x = 1 and x = 2.
Solution:
Hence f is not differentiable at x = 2