Differentiation Class 12 OP Malhotra Exe 8D ISC Maths Solutions Ch-8. In this article you would learn differentiation of logarithmic and exponential functions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

Differentiation Class 12 OP Malhotra Exe 8D ISC Maths Solutions Ch-8

Board	ISC
Publications	S Chand
Subject	Maths
Class	12th
Chapter-8	Differentiation
Writer	OP Malhotra
Exe-8(D)	differentiation of logarithmic and exponential functions

differentiation of logarithmic and exponential functions

Differentiation Class 12 OP Malhotra Exe 8D ISC Maths Solutions Ch-8

Que-1:
(i) log cos x
(ii) log sin x
(iii) cos (log x)
(iv) 1/log cosx
(v) x log x – x

Sol: (i) Let y = log cos x
Diff both sides w.r.t. x ; we have
dy/dx= 1/cosx d/dx cosx = -sinx/cosx = – tanx

(ii) Let y = log sin x
Diff both sides w.r.t. x ; we have
dy/dx=1/sinx d/dx sinx = cosx/sinx =cotx

(iii) Let y = cos (log x)
DifF both sides w.r.t. x ; we have
dy/dx=d/dx cos(logx)=-sin(logx)d/dx logx
=-sin(logx)/x

(iv) 1/log cosx
Diff both sides w.r.t. x ; we have
dy/dx=d/dx(1/log cosx)
=-1/(log cosx)²d/dx log cosx
=-1/(log cosx)² 1/cosx (-sinx)
=tanx/(log cosx)²

v) Let y = x log x – x
Diff both sides w.r.t. x ; we have
dy/dx=x.d/dx logx + logx d/dx x -d/dx x
= x.1/x + log x. 1-1 = 1 + log x – 1 = log x

Que-2:
(i) log(x^(3/5)
(ii) log (3 – 7x)
(iii) log x³
(iv) log √x
(v) sinx/logx

Sol: (i) = logx^3/5
= 3/5log x [∵ log a^b = b log a]
Diff both sides w.r.t. x ; we have
dy/dx=3/5x

(ii) Let y = log (3 – 7x);
Diff both sides w.r.t. x ; we have
dy/dx=d/dx log(3-7x)
= 1/3-7x d/dx (3-7x) = -7/(3-7x)

(iii) Let y = log x³ = 3 log x
Diff both sides w.r.t. x ; we have
dy/dx=3/x

(iv) Let y = log √x = log x^1/2 =1/2 log x
Diff both sides w.r.t. x ; we have
dy/dx=1/2x

(v) Let y = sinx/logx

Que-3: log (cosecx – cotx)

Sol:

Que-4: sin(log cos x)

Sol: Let y = sin (log cos x)
Diff both sides w.r.t. x ; we have
dy/dx=cos(log cosx) d/dx log cosx
cosx(log cosx) 1/cosx d/dx cosx
= -sinx/cosx cos(log cosx)
= – tan x cos(log cos x)

Que-5: log (log x)

Sol: Let y = log(log x)
Diff both sides w.r.t. x ; we have
dy/dx= 1/logx d/dx (logx) = 1/xlogx

Que-6: logx/(1+logx)

Sol:

Que-7: sin (log x) – log sin x

Sol: Let y = sin (log x) – log sin x
Diff both sides w.r.t. x ; we have
∴ dy/dx= d/dx sin(logx) – d/dx log(sinx)
= cos (log x).1/x – (1/sinx).cosx
= cos(log x)/x – cot x

Que-8: log((1-x²)/(1+x²))

Sol:

Que-9: log√(x-1/x+1)

Sol:

Que-10: sin [sin (log 3x)]

Sol: Let y = sin [sin (log 3x)]
Diff both sides w.r.t. x ; we have
∴ dy/dx= cos(sin(log 3x)) d/dx sin(log 3x)
= cos(sin (log 3x)) cos(log 3x) d/dx (log 3x)
= cos(sin(log 3x)) cos(log3x) 1/3x × 3
= (cos [sin (log 3x)] cos(log 3x))/x

Que-11: log cos √x

Sol:

Que-12: cos (log x)²

Sol: Let y = cos (log x)²
Diff both sides w.r.t. x ; we have
∴ dy/dx = -sin(log x)² d/dx (log x)²
= -sin(log x)² 2(log x) d/dx(log x)
= -2sin(log x)² logx / x

Que-13: log(√tanx)

Sol:

Que-14: log(x+√(1+x²))

Sol:

Que-15: log(x-√(x²-a²))

Sol:

Que-16: sec x. tan x + log tan (π/4 + x/2)

Sol: sec x. tan x + log tan (π/4 + x/2) ; Diff both sides w.r.t. x ; we have

Que-17: log(√(1+sinx/1-sinx))

Sol:

Que-18: log(1-x/1+x)^1/3

Sol:

Que-19: (ln ln x)²

Sol:

Que-20: log(sec x/2 + tan x/2)

Sol:

Que-21: log [sin (log x)]

Sol: Let y = log [sin (log x)]
Diff both sides w.r.t. x ; we have
dy/dx = 1/sin(log x) d/dx sin(log x) = cos(log x)/sin(log x) d/dx(log x) = cot(log x)/x

Que-22: log[log(sin√(x²+1))]

Sol:

Que-23: log(√a+x + √a-x / √a+x – √a-x)

Sol:

Que-24: If y = √(x²+1) – log(1/x + √(1+1/x²) , find dy/dx

Sol:

–: End of Differentiation Class 12 OP Malhotra Exe-8D ISC Math Ch-8 Solution :–

Return to :- OP Malhotra ISC Class-12 S Chand Publication Maths Solutions

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