# Class 12 RD Sharma Solutions – Chapter 11 Differentiation – Exercise 11.5 | Set 2

### Question 21. Find dy/dx when .

**Solution:**

We have,

=>

=>

On taking log of both the sides, we get,

=>

=>

=>

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

=>

=>

=>

=>

=>

**Que**stion 22. Find dy/dx when .

**Solution:**

We have,

=>

=>

On taking log of both the sides, we get,

=>

=>

=>

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

=>

=>

=>

=>

=>

### Question 23. Find dy/dx when y = e^{3x} sin 4x 2^{x}.

**Solution:**

We have

=> y = e

^{3x}sin 4x 2^{x}.On taking log of both the sides, we get,

=> log y = log (e

^{3x}sin 4x 2^{x})=> log y = log e

^{3x}+ log (sin 4x) + log 2^{x}=> log y = 3x log e + log (sin 4x) + x log 2

=> log y = 3x + log (sin 4x) + x log 2

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

=>

=>

=>

=>

=>

### Question 24. Find dy/dx when y = sin x sin 2x sin 3x sin 4x.

**Solution:**

We have,

=> y = sin x sin 2x sin 3x sin 4x

On taking log of both the sides, we get,

=> log y = log (sin x sin 2x sin 3x sin 4x)

=> log y = log sin x + log sin 2x + log sin 3x + log sin 4x

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

=>

=>

=> = cotx + 2cot2x + 3cot3x + 4cot4x

=> = y(cotx + 2cot2x + 3cot3x + 4cot4x)

=> = (sinxsin2x sin3xsin4x)(cotx + 2cot2x + 3cot3x + 4cot4x)

### Question 25. Find dy/dx when y = x^{sin x} + (sin x)^{x}.

**Solution:**

We have,

=> y = x

^{sin x}+ (sin x)^{x}.Let u = x

^{sin x}and v = (sin x)^{x}. Therefore, y = u + v.Now, u = x

^{sin x}On taking log of both the sides, we get,

=> log u = log x

^{sin x}=> log u = sin x log x

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

=>

=>

=>

=>

=>

Also, v = (sin x)

^{x}On taking log of both the sides, we get,

=> log v = log (sin x)

^{x}=> log v = x log sin x

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

=>

=>

=>

=>

=>

Now we have, y = u + v.

=>

=>

### Question 26. Find dy/dx when y = (sin x)^{cos x} + (cos x)^{sin x}.

**Solution:**

We have,

=> y = (sin x)

^{cos x}+ (cos x)^{sin x}=>

=>

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

=>

=>

=> = (sinx)

^{cosx}[cosxcotx – sinxlog(sinx)] + (cosx)^{sinx}[-tanxsinx + cosxlog(cosx)]=> = (sinx)

^{cosx}[cosxcotx – sinxlog(sinx)] + (cosx)^{sinx}[cosxlog(cosx) – tanxsinx]

### Question 27. Find dy/dx when y = (tan x)^{cot x} + (cot x)^{tan x}.

**Solution:**

We have,

=> y = (tan x)

^{cot x}+ (cot x)^{tan x}=>

=>

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

=>

=>

=>

=> = (tanx)

^{cotx}[cosec^{2}x – log(tanx)(cosec^{2}x)] + (cotx)^{tanx}[-sec^{2}x + log(cotx)(sec^{2}x)]=> = (tanx)

^{cotx}[cosec^{2}x – cosec^{2}xlog(tanx)] + (cotx)^{tanx}[sec^{2}xlog(cotx) – sec^{2}x]

### Question 28. Find dy/dx when y = (sin x)^{x} + sin^{−1 }√x.

**Solution:**

We have,

=> y = (sin x)

^{x}+ sin^{−1}√x=>

=>

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

=>

=>

=>

=>

### Question 29. Find dy/dx when

### (i) y = x^{cos x} + (sin x)^{tan x}

**Solution:**

We have,

=> y = x

^{cos x}+ (sin x)^{tan x}=>

=>

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

=>

=>

=>

=>

### (ii) y = x^{x} + (sin x)^{x}

**Solution:**

We have,

=> y = x

^{x}+ (sin x)^{x}=>

=>

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

=>

=>

=>

=>

### Question 30. Find dy/dx when y = (tan x)^{log x} + cos^{2} (π/4).

**Solution:**

We have,

=> y = (tan x)

^{log x}+ cos^{2}(π/4)=>

=>

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

=>

=>

=>

=>

### Question 31. Find dy/dx when .

**Solution:**

We have,

=>

=>

=>

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

=>

=>

=>

=>

=>

### Question 32. Find dy/dx when y = (log x)^{x}+ x^{logx}.

**Solution:**

We have,

=> y = (log x)

^{x}+ x^{logx}Let u = (log x)

^{x}and v = x^{logx}. Therefore, y = u + v.Now, u = (log x)

^{x}On taking log of both the sides, we get,

=> log u = log (log x)

^{x}=> log u = x log (log x)

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

=>

=>

=>

=>

=>

=>

=>

Also, v = x

^{logx}On taking log of both the sides, we get,

=> log v = log x

^{logx}=> log v = log x (log x)

=> log v = (log x)

^{2}On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

Now, y = u + v

=>

=>

### Question 33. If x^{13}y^{7} = (x+y)^{20}, prove that .

**Solution:**

We have,

=> x

^{13}y^{7}= (x+y)^{20}On taking log of both the sides, we get,

=> log x

^{13}y^{7 }= log (x+y)^{20}=> log x

^{13 }+ log y^{7 }= log (x+y)^{20}=> 13 log x + 7 log y = 20 log (x+y)

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

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 34. If x^{16}y^{9} = (x^{2} + y)^{17}, prove that .

**Solution:**

We have,

=> x

^{16}y^{9}= (x^{2 }+ y)^{17}On taking log of both the sides, we get,

=> log x

^{16}y^{9}= log (x^{2}+ y)^{17}=> log x

^{16}+ log y^{9}= log (x^{2}+y)^{17}=> 16 log x + 9 log y = 17 log (x

^{2}+ y)On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 35. If y = sin x^{x}, prove that .

**Solution:**

We have,

=> y = sin x

^{x}Let u = x

^{x}. Now y = sin u.On taking log of both the sides, we get,

=> log u = log x

^{x}=> log u = x log x

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

=>

=>

=>

=>

Now, y = sin u

=>

=>

=>

Hence proved.

### Question 36. If x^{x} + y^{x} = 1, prove that .

**Solution:**

We have,

=> x

^{x}+ y^{x}= 1=>

=>

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

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 37. If x^{y} × y^{x} = 1, prove that .

**Solution:**

We have,

=> x

^{y}× y^{x}= 1On taking log of both the sides, we get,

=> log (x

^{y }× y^{x}) = log 1=> log x

^{y }+ log y^{x}= log 1=> y log x + x log y = log 1

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

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 38. If x^{y }+ y^{x} = (x+y)^{x+y}, find dy/dx.

**Solution:**

We have,

=> x

^{y}+ y^{x}= (x+y)^{x+y}=>

=>

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

=>

=>

=>

=>

=>

### Question 39. If x^{m} y^{n} = 1, prove that .

**Solution:**

We have,

=> x

^{m}y^{n}= 1On taking log of both the sides, we get,

=> log (x

^{m}y^{n})= log 1=> log x

^{m}+ log y^{n}= log 1=> m log x + n log y = log 1

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

=>

=>

=>

=>

Hence proved.

### Question 40. If y^{x} = e^{y−x}, prove that .

**Solution:**

We have,

=> y

^{x}= e^{y−x}On taking log of both the sides, we get,

=> log y

^{x}= log e^{y−x}=> x log y = (y − x) log e

=> x log y = y − x

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

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

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