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

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 e^x + e^y = e^x+y, prove that .

Solution:

We have,

=> e^x + e^y = e^x+y

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

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

Hence proved.

Question 44. If e^y = y^x, prove that .

Solution:

We have,

=> e^y = y^x

On taking log of both the sides, we get,

=> log e^y = log y^x

=> 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 e^x+y − x = 0, prove that .

Solution:

We have,

=> e^x+y − x = 0

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

=> 

=> 

Now, we know e^x+y − x = 0

=> e^x+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.

Question 51. Find the derivative of the function f(x) given by,

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 = e^x-y.

Solution:

We have,

=> xy = e^x-y

On taking log of both the sides, we get,

=> log xy = log e^x-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 y^x + x^y + x^x = a^b.

Solution:

We have,

=> y^x + x^y + x^x = a^b

=> 

=> 

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,

=>