# 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 = 0On differentiating both sides with respect to x, we get,

=>

=>

Now, we know e

^{x+y}− x = 0=> e

^{x+y}= xSo, 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 + yOn 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+x^{2}) (1+x^{4}) (1+x^{8}) and hence find f'(1).

**Solution:**

Here we are given,

=> f(x) = (1+x) (1+x

^{2}) (1+x^{4}) (1+x^{8})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,

=>