# NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

These NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 Questions and Answers are prepared by our highly skilled subject experts.

## NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.7

Question 1.
x² + 3x + 2
Solution:
Let y = x² + 3x + 2
Differentiating both sides w.r.t. x
$$\frac { dy }{ dx }$$ = 2x + 3 and $$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$$ = 2

Question 2.
x20 = y
Solution:
Let y = x20
Differentiating both sides w.r.t. x
$$\frac { dy }{ dx } ={ 20 }x^{ 19 }\quad ⇒\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =20\times { 19x }^{ 18 }={ 380 }x^{ 18 }\qquad$$

Question 3.
x.cos x = y(say)
Solution:
Let y = x.cos x
Differentiating both sides w.r.t. x
$$\frac { dy }{ dx }$$ = x(- sinx) + cosx.1 = – xsinx + cosx
$$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$$ = – xcosx – sinx – sinx = – xcosx – 2sinx

Question 4.
log x = y (say)
Solution:
Let y = log x
Differentiating both sides w.r.t. x
$$\frac { dy }{ dx } =\frac { 1 }{ x } ⇒ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =-\frac { 1 }{ { x }^{ 2 } }$$

Question 5.
x³ log x = y (say)
Solution:
Let y = x³ log x
Differentiating both sides w.r.t. x
$$⇒\frac { dy }{ dx } ={ x }^{ 3 }.\frac { 1 }{ x } +logx\times { 3x }^{ 2 }={ x }^{ 2 }+{ 3x }^{ 2 }logx$$
$$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =2x+{ 3x }^{ 2 }.\frac { 1 }{ x } +logx.6x=x(5+6logx)$$

Question 6.
ex sin 5x = y
Solution:

Question 7.
e6x cos3x
Solution:
Let y = e6x cos3x
Differentiating both sides w.r.t. x

Question 8.
tan-1 x
Solution:
Let y = tan-1 x
Differentiating both sides w.r.t. x
$$\frac { dy }{ dx } =\frac { 1 }{ 1+{ x }^{ 2 } } ⇒\frac { { d }^{ 2y } }{ { dx }^{ 2 } } =\frac { -2x }{ { ({ 1+x }^{ 2 }) }^{ 2 } }$$

Question 9.
log(logx)
Solution:
Let y = log(logx)
Differentiating both sides w.r.t. x

Question 10.
sin(log x)
Solution:
Let y = sin(log x)
Differentiating both sides w.r.t. x

Question 11.
If y = 5 cos x – 3 sin x, prove that $$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +y=0$$
Solution:
Let y = 5 cos x – 3 sin x
Differentiating both sides w.r.t. x
$$\frac { dy }{ dx }$$ = – 5sinx – 3cosx
$$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$$ = – 5cosx +3sinx = – y
$$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$$ + y = 0
Hence proved

Question 12.
If y = cos-1 x, Find $$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$$ in terms of y alone.
Solution:
Let y = cos-1 x
Differentiating both sides w.r.t. x
$$\frac { dy }{ dx } = – { \left( { 1-x }^{ 2 } \right) }^{ -\frac { 1 }{ 2 } }$$
$$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =\frac { -cosy }{ { \left( { sin }^{ 2 }y \right) }^{ \frac { 3 }{ 2 } } } = -coty\quad { cosec }^{ 2 }y$$

Question 13.
If y = 3 cos (log x) + 4 sin (log x), show that
$${ x }^{ 2 }{ y }_{ 2 }+{ xy }_{ 1 }$$ + y = 0
Solution:
Let y = 3 cos (log x) + 4 sin (log x)
Differentiating both sides w.r.t. x

Question 14.
If A = Aemx + Benx, show that $$\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y$$ = 0
Solution:
Let y = Aemx + Benx
Differentiating both sides w.r.t. x
$$\frac { d }{ dx }$$ = Aemx + Benx
Differentiating both sides w.r.t. x
$$\frac{d^{2} y}{d x^{2}}$$ = Aemx(m) + Benx(n)
$$\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y$$
= Aemx . (m²) + Benx(n²) – (m + n) [Aemx(m) + Benx(n)] + mny
= Am²emx + Bn²enx – [Am²emx – Bn²enx +mny
= – mn[Aemx + Benx] + mny
= – mny + mny = 0 = RHS

Question 15.
If y = 500e7x + 600e-7x, show that $$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$$ = 49y.
Solution:
Let y = 500e7x + 600e-7x
Differentiating both sides w.r.t. x

Question 16.
If ey(x+1) = 1, show that $$\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}$$
Solution:
Let y = ey(x+1)
Differentiating both sides w.r.t. x

Question 17.
If y = (tan-1 x)² show that (x² + 1)²y2 + 2x(x² + 1)y1 = 2
Solution:
Let y = (tan-1 x)²
Differentiating both sides w.r.t. x

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