Differentiation Class 12 OP Malhotra Exe-8K ISC Maths Solutions

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Differentiation Class 12 OP Malhotra Exe-8K ISC Maths Solutions Ch-8 Solutions. In this article you would learn about logarithmic differentiation. Step by step solutions of latest textbook has been given as latest syllabus. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

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

Board	ISC
Publications	S Chand
Subject	Maths
Class	12th
Chapter-8	Differentiation
Writer	OP Malhotra
Exe-8(K)	logarithmic differentiation

Logarithmic Differentiation

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

Que-1: (x² + 2)³ (1 – x³)⁴

Sol: Let y – (x² + 2)³ (1 – x³)⁴
Taking logarithm on both sides, we have
log y – log (x² + 2)³ (1 – x³)⁴
⇒ log y = 3 log(x² + 2) + 4log(1 – x³)
[∵ log ab = log a + log b and log a^b = b log a]
Diff. both sides w.r.t. x; we have

Que-2: x(1-x²)²/(1+x²)^1/2

Sol: Let y = x(1-x²)²/(1+x²)^1/2
Taking logarithm on both sides, we have

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Que-3: (x+1)²√(x-1)/(x+4)³) e^x

Sol:

Que-4: √((x-1)(x-2)(x-3)(x-4))

Sol:

Que-5: (x-a)(x-b)/(x-p)(x-q)

Sol:

Que-6: 2(x-sinx)^1/2/√x

Sol: Let y =2(x-sinx)^1/2/√x
Taking logarithm on both sides, we have
log y = log 2 + 3/2log (x – sin x) – 1/2 log x
Diff. both sides w.r.t. x; we have
1/y dy/dx = [3/2 1/(x-sinx)[1-cosx] – 1/2x]
Thus, dy/dx = 2(x-sinx)²/√x [3/2 (1-cosx/x-sinx) -1/2x]

Que-7: (i) x^1/x
(ii) x^√x(iii) (1/x)^x

Sol: (i) Let y = x^1/x ;
Taking logarithm on both sides, we have
log y = 1/x log x ;
Diff. both sides w.r.t. x; we have
1/y dy/dx = 1/x . 1/x + logx (-1/x²)
⇒ dy/dx = (x^1/x)/x² (1-logx)

(ii) Let y =x^√x;
Taking logarithm on both sides, we have
log y = log x^√x = √x log x ;
Diff. both sides w.r.t. x; we have
1/y dy/dx = √x/x + log x; 1/2√x
⇒ dy/dx = y[1/√x + log x/2√x]
= x^√x[1/√x + log x/2√x]

(iii) Let (1/x)^x;
Taking logarithm on both sides, we have
log y = log ((1/x)^x) = x log (1/x)
= x(- log x) = – x log x
Diff. both sides w.r.t. x; we have
1/y dy/dx = – [x.1/x + log x . 1];
⇒ dy/dx = – y(1 + log x)
= – (1/x)^x (1+ log x)

Que-8: (sin x)^x

Sol: Let y = (sin x)^x;
Taking logarithm on both sides, we have
log y = x log sin x ;
Diff. both sides w.r.t. x; we have
1/y dy/dx = x/sin x . cos x + log sinx
⇒ dy/dx = y[xcotx + log sin x]
= (sin x)^x[x cot x + log sin x]

Que-9: x^{sin x}

Sol: Let y = x^{sin x} ;
Taking logarithm on both sides, we have
log y = log x^{sin x} = sin x . log x ;
Diff. both sides w.r.t. x; we have
1/y dy/dx = sin x /x cos x + logxcosx
⇒ dy/dx = y[sin x /x + (log x)cos x]
= (x^{sin x})[sin x /x + (log x)cos x]

Que-10: (sin x)^{tan x}

Sol: Let y = (sin x)^{tan x} ;
Taking logarithm on both sides, we have
log y = log (sin x)^{tan x}
= tan x . log sin x ;
Diff. both sides w.r.t. x; we have
1/y dy/dx = tan x . 1/sin x cos x + log sin x sec² x
⇒ dy/dx = y[1 + sec² x log sin x]
= (sin x)^{tan x}[1 + sec² x log sin x]

Que-11: (tan x)^{log x}

Sol: Let y = (tan x)^{log x} ;
Taking logarithm on both sides, we have
log y = log (tan x)^{log x} – log x . log tan x ;
Diff. both sides w.r.t. x; we have
1/y dy/dx = log x 1/tan x sec²x + log tan x . 1/x ⇒ dy/dx = y[log x/sin x.cos x +log tan x/ x]
⇒ dy/dx = ((tan x)^{log x}) [log x /sin x cos x + log tan x/ x]

Que-12: x^{log x}

Sol: Let y = x^{log x} ;
Taking logarithm on both sides, we have
log y = log x^{log x} = log x . log x = (log x)² ;
Diff. both sides w.r.t. x; we have
1/y dy/dx = 2 log x . 1/x ⇒ dy/dx = y[2log x/ x] = x^{log x}. 2log x/ x

Que-13: (tan x)^{cos x}

Sol:

Que-14: (log x)^x

Sol: Let y = (log x)^x ;
Taking logarithm on both sides, we have
log y = log(log x)^x = x log(log x) ;
Diff. both sides w.r.t. x; we have
1/y dy/dx = log(log x) . 1 + x . 1/log x . 1/x
⇒ dy/dx = y[log(log x) + 1/log x] ⇒ dy/dx = (log x)^x[log(log x) + 1/log x]

Que-15: (1+1/x)^x

Sol:

Que-16: x^x√x

Sol:

Que-17: (i) cos x^x(ii) sin (x^x)
(iii) If y = (tan πx/4)^4/πx, find dy/dx at x = 1.

Sol: (i) Let y = cos x^x ;
Diff. both sides w.r.t. x; we have

Que-18: Find dy/dx if
(i) y = log (x^x + cosec² x)
(ii) y = e^sin²x(2tan^-1√1-x/1+x)
(iii) x^y = y^x
(iv) (cos x)^y = (siny)^x

Sol: (i) Given y = log (x^x + cosec² x); Diff both sides w.r.t. x; we have

(ii) Given y = e^sin²x(2tan^-1√1-x/1+x)
Taking logarithm on both sides; we have

(iii) x^y = y^x;
Taking logarithm on both sides y log x = x log y ;
Diff. both sides w.r.t. x; we have
y/x + log x dy/dx = x/y dy/dx + log y . 1
⇒ (log x – x/y) dy/dx = log y – x/y
⇒ dy/dx = y(xlogy – y)/x(ylogx – x)

(iv) Given (cos x)^y = (sin y)^x;
Taking logarithm on both sides; we have
log (cos x)^y = log (sin y)^x
y log cos x = x log sin y;
Diff. both sides w.r.t. x; we have

Que-19: If y = e^x-y, show that dy/dx = y/(1+y) .

Sol: Given y = e^x-y ;
Taking logarithm on both sides; we have
log y = log e^x-y = x – y ;
Diff. both sides w.r.t. x; we have
⇒ 1/y dy/dx = 1 – dy/dx
⇒ (1/y + 1) dy/dx = 1 ⇒ (1+y/y) dy/dx = 1
⇒ dy/dx = y/1+y

Que-20: If x^myⁿ = (x + y)^m+n, prove that dy/dx = y/x.

Sol: Given x^myⁿ = (x + y)^m+n ;
Taking logarithm on both sides; we have
log x^m + log yⁿ = log (x + y)^m+n
⇒ m log x + n log y = (m + n) log(x + y) [∵ log ab = log a + log b & log a^b = b log a]
Diff. both sides w.r.t. x; we have

Que-21: If y = x^{x^(x………∞)}, prove that dy/dx = y²/x(1-ylogx)

Sol:

Que-22: If y = √x^{√x^(√x…..∞)} find dy/dx

Sol:

Que-23: If y = a^{x^(}a^x^…..∞)

Sol: Given y = a^{x^(y)}
Taking logarithm on both sides; we have
log y = log a^{x^(y)}= log a A
gain taking logarithm on both sides, we have
log log y = log(x^y log a) = y log x + log log a;
Diff. both sides w.r.t. x; we have

Que-24: If y = x^y, prove that x dy/dx = y²/(1-ylogx).

Sol: Given y = x^{x^(x…..∞)} = x^y
Taking logarithm on both sides; we have
log y = y log x ;
Diff. both sides w.r.t. x; we have

Que-25: Find dy/dx when x^y + y^x = c.

Sol: Given x^y + y^x = c ⇒ u + v = c …(1)
where u = x^y …(2)
& v = y^x …(3)
Diff. eqn. (1) both sides w.r.t. x; we get
du/dx + dy/dx = 0… (4)
Taking logarithm on both sides of eqn. (2); we have
log u = y log x; diff. w.r.t. x, we have
1/u du/dx = y/x + log x dy/dx
⇒ du/dx = x^y[y/x + log x dy/dx]… (5)
Taking logaritum on both sides of eqn.(3); we have
log v = x log y
Diff. both sides w.r.t. x; we have

Que-26: If x^y = e^y-x, prove that dy/dx = 2-log x/(1-log x)².

Sol: Given x^y = e^y-x;
Taking logarithm on both sides; we have
y log x = (y – x) log e – y – x
⇒ y(1 – log x) = x

Que-27: Differentiate (sin x)^x w.r.t. x².

Sol: Let y = (sin x)^x …(1)
& z = x² …(2)
Taking logarithm on both sides of eqn. (1); we have
log y = x log sin x ; diff. w.r.t. x, we have

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

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Differentiation Class 12 OP Malhotra Exe-8K ISC Maths Solutions

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

Logarithmic Differentiation

Que-1: (x² + 2)³ (1 – x³)4

Que-2: x(1-x²)²/(1+x²)1/2

Que-3: (x+1)²√(x-1)/(x+4)³) ex

Que-4: √((x-1)(x-2)(x-3)(x-4))

Que-5: (x-a)(x-b)/(x-p)(x-q)

Que-6: 2(x-sinx)1/2/√x

Que-7: (i) x1/x (ii) x√x (iii) (1/x)x

Que-8: (sin x)x

Que-9: xsin x

Que-10: (sin x)tan x

Que-11: (tan x)log x

Que-12: xlog x

Que-13: (tan x)cos x

Que-14: (log x)x

Que-15: (1+1/x)x

Que-16: xx√x

Que-17: (i) cos xx (ii) sin (xx) (iii) If y = (tan πx/4)4/πx, find dy/dx at x = 1.

Que-18: Find dy/dx if (i) y = log (xx + cosec² x) (ii) y = esin²x(2tan-1√1-x/1+x) (iii) xy = yx (iv) (cos x)y = (siny)x

Que-19: If y = ex-y, show that dy/dx = y/(1+y) .

Que-20: If xmyn = (x + y)m+n, prove that dy/dx = y/x.

Que-21: If y = xx^(x………∞), prove that dy/dx = y²/x(1-ylogx)

Que-22: If y = √x√x^(√x…..∞) find dy/dx

Que-23: If y = ax^(ax…..∞)

Que-24: If y = xy, prove that x dy/dx = y²/(1-ylogx).

Que-25: Find dy/dx when xy + yx = c.

Que-26: If xy = ey-x, prove that dy/dx = 2-log x/(1-log x)².

Que-27: Differentiate (sin x)x w.r.t. x².

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Que-1: (x² + 2)³ (1 – x³)⁴

Que-2: x(1-x²)²/(1+x²)^1/2

Que-3: (x+1)²√(x-1)/(x+4)³) e^x

Que-6: 2(x-sinx)^1/2/√x

Que-7: (i) x^1/x
(ii) x^√x(iii) (1/x)^x

Que-8: (sin x)^x

Que-9: x^{sin x}

Que-10: (sin x)^{tan x}

Que-11: (tan x)^{log x}

Que-12: x^{log x}

Que-13: (tan x)^{cos x}

Que-14: (log x)^x

Que-15: (1+1/x)^x

Que-16: x^x√x

Que-17: (i) cos x^x(ii) sin (x^x)
(iii) If y = (tan πx/4)^4/πx, find dy/dx at x = 1.

Que-18: Find dy/dx if
(i) y = log (x^x + cosec² x)
(ii) y = e^sin²x(2tan^-1√1-x/1+x)
(iii) x^y = y^x
(iv) (cos x)^y = (siny)^x

Que-19: If y = e^x-y, show that dy/dx = y/(1+y) .

Que-20: If x^myⁿ = (x + y)^m+n, prove that dy/dx = y/x.

Que-21: If y = x^{x^(x………∞)}, prove that dy/dx = y²/x(1-ylogx)

Que-22: If y = √x^{√x^(√x…..∞)} find dy/dx

Que-23: If y = a^{x^(}a^x^…..∞)

Que-24: If y = x^y, prove that x dy/dx = y²/(1-ylogx).

Que-25: Find dy/dx when x^y + y^x = c.

Que-26: If x^y = e^y-x, prove that dy/dx = 2-log x/(1-log x)².

Que-27: Differentiate (sin x)^x w.r.t. x².