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# Class 12 RD Sharma Solutions – Chapter 11 Differentiation – Exercise 11.5 | Set 3

• Last Updated : 26 May, 2021

### Question 41. If (sin x)y = (cos y)x, prove that .

Solution:

We have,

=> (sin x)y = (cos y)x

On taking log of both the sides, we get,

=> log (sin x)y = log (cos y)x

=> y log (sin x) = x log (cos y)

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

Hence proved.

### Question 42. If (cos x)y = (tan y)x, prove that \frac{dy}{dx}=\frac{log tany+ytanx}{logcosx-xsecycosecy}     .

Solution:

We have, (cos x)y = (tan y)x

On taking log of both the sides, we get,

=> log (cos x)y = log (tan y)x

=> y log (cos x) = x log (tan y)

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 43. If ex + ey = ex+y, prove that .

Solution:

We have,

=> ex + ey = ex+y

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 44. If ey = yx, prove that .

Solution:

We have,

=> ey = yx

On taking log of both the sides, we get,

=> log ey = log yx

=> y log e = x log y

=> y = x log y

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 45. If ex+y − x = 0, prove that .

Solution:

We have,

=> ex+y − x = 0

On differentiating both sides with respect to x, we get,

=>

=>

Now, we know ex+y − x = 0

=> ex+y = x

So, we get,

=>

=>

=>

=>

=>

Hence proved.

### Question 46. If y = x sin (a+y), prove that .

Solution:

We have,

=> y = x sin (a+y)

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

Now we know, y = x sin (a+y)

=>

So, we get,

=>

=>

=>

=>

Hence proved.

### Question 47. If x sin (a+y) + sin a cos (a+y) = 0, prove that .

Solution:

We have,

=> x sin (a+y) + sin a cos (a+y) = 0

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

Now we know, x sin (a+y) + sin a cos (a+y) = 0

=>

So, we get,

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 48. If (sin x)y = x + y, prove that .

Solution:

We have,

=> (sin x)y = x + y

On taking log of both the sides, we get,

=> log (sin x)y = log (x + y)

=> y log sin x = log (x + y)

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 49. If xy log (x+y) = 1, prove that .

Solution:

We have,

=> xy log (x+y) = 1

On differentiating both sides with respect to x, we get,

=>

=>

=>

Now, we know, xy log (x+y) = 1.

=>

So, we get,

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 50. If y = x sin y, prove that .

Solution:

We have,

=> y = x sin y

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

Now, we know y = x sin y

=>

So, we get,

=>

=>

=>

Hence proved.

### f(x) = (1+x) (1+x2) (1+x4) (1+x8) and hence find f'(1).

Solution:

Here we are given,

=> f(x) = (1+x) (1+x2) (1+x4) (1+x8)

On differentiating both sides with respect to x, we get,

=>

=>

Now, the value of f'(x) at 1 is,

=> f'(1) = (1 + 1) (1 + 1) (1 + 1) (8) + (1 + 1) (1 + 1) (1 + 1) (4) + (1 + 1) (1 + 1) (1 + 1) (2) + (1 + 1) (1 + 1) (1 + 1) (1)

=> f'(1) = (2) (2) (2) (8) + (2) (2) (2) (4) + (2) (2) (2) (2) + (2) (2) (2) (1)

=> f'(1) = 64 + 32 + 16 + 8

=> f'(1) = 120

Therefore, the value of f'(1) is 120.

### Question 52. If , find .

Solution:

We are given,

=>

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

### Question 53. If y = (sin x − cos x)sin x−cos x, π/4 < x < 3π/4, find .

Solution:

We have,

=> y = (sin x − cos x)sin x−cos x

On taking log of both the sides, we get,

=> log y = log (sin x − cos x)sin x−cos x

=> log y = (sin x − cos x) log (sin x−cos x)

On differentiating both sides with respect to x, we get,

=>

=> = (1)(cosx + sinx) + (cosx + sinx)log (sin x − cos x)

=> = cosx + sinx + (cosx + sinx)log (sin x − cos x)

=> = (cosx + sinx)(1 + log (sin x − cos x))

=> = y(cosx + sinx)(1 + log (sin x − cos x))

=> = (sinx – cosx)sinx-cosx(cosx + sinx)(1 + log (sin x − cos x))

### Question 54. Find dy/dx of function xy = ex-y.

Solution:

We have,

=> xy = ex-y

On taking log of both the sides, we get,

=> log xy = log ex-y

=> log x + log y = (x − y) log e

=> log x + log y = x − y

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

### Question 55. Find dy/dx of function yx + xy + xx = ab.

Solution:

We have,

=> yx + xy + xx = ab

=>

=>

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

### Question 56. If (cos x)y = (cos y)x, find dy/dx.

Solution:

We have,

=> (cos x)y = (cos y)x

On taking log of both the sides, we get,

=> log (cos x)y = log (cos y)x

=> y log (cos x) = x log (cos y)

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

### Question 57. If cos y = x cos (a+y), where cos a ≠ ±1, prove that .

Solution:

We have,

=> cos y = x cos (a+y)

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 58. If , prove that .

Solution:

We have,

=>

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 59. If , prove that .

Solution:

We have,

=>

On taking log of both the sides, we get,

=> log x = log

=>

=>

=>

On differentiating both sides with respect to x, we get,

=>

=>

We know,

=>

So, we get,

=>

=>

=>

=>

=>

Hence proved.

### Question 60. If , find dy/dx.

Solution:

We have,

=>

=>

=>

On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

### Question 61. If , find dy/dx.

Solution:

We are given,

=>

Now we know,

If  then,

In the given expression, we have 1/x instead of x.

So, using the above theorem, we get,

=>

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