# Class 12 RD Sharma Solutions – Chapter 11 Differentiation – Exercise 11.5 | Set 1

### Question 1. Differentiate y = x1/x with respect to x.

Solution:

We have,

=> y = x1/x

On taking log of both the sides, we get,

=> log y = log x1/x

=> log y = (1/x) (log x)

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

=>

=>

=>

=>

=>

=>

### Question 2. Differentiate y = xsin x with respect to x.

Solution:

We have,

=> y = xsin x

On taking log of both the sides, we get,

=> log y = log xsin x

=> log y = sin x log x

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

=>

=>

=>

=>

=>

### Question 3. Differentiate y = (1 + cos x)x with respect to x.

Solution:

We have,

=> y = (1 + cos x)x

On taking log of both the sides, we get,

=> log y = log (1 + cos x)x

=> log y = x log (1 + cos x)

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

=>

=>

=>

=>

=>

=>

### Question 4. Differentiate  with respect to x.

Solution:

We have,

=>

On taking log of both the sides, we get,

=> log y = log

=> log y = cosâˆ’1 x log x

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

=>

=>

=>

=>

=>

### Question 5. Differentiate y = (log x)x with respect to x.

Solution:

We have,

=> y = (log x)x

On taking log of both the sides, we get,

=> log y = log (log x)x

=> log y = x log (log x)

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

=>

=>

=>

=>

=>

### Question 6. Differentiate y = (log x)cos x with respect to x.

Solution:

We have,

=> y = (log x)cos x

On taking log of both the sides, we get,

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

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

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

=>

=>

=>

=>

=>

### Question 7. Differentiate y = (sin x)cos x with respect to x.

Solution:

We have,

=> y = (sin x)cos x

On taking log of both the sides, we get,

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

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

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

=>

=>

=>

=>

=>

=>

### Question 8. Differentiate y = ex log x with respect to x.

Solution:

We have,

=> y=ex log x

=> y =

=> y = xx

On taking log of both the sides, we get,

=> log y = log xx

=> log y = x log x

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

=>

=>

=>

=>

=>

### Question 9. Differentiate y = (sin x)log x with respect to x.

Solution:

We have,

=> y = (sin x)log x

On taking log of both the sides, we get,

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

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

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

=>

=>

=>

=>

=>

### Question 10. Differentiate y = 10log sin x with respect to x.

Solution:

We have,

=> y = 10log sin x

On taking log of both the sides, we get,

=> log y = log 10log sin x

=> log y = log (sin x) log 10

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

=>

=>

=>

=>

=>

=>

### Question 11. Differentiate y = (log x)log x with respect to x.

Solution:

We have,

=> y = (log x)log x

On taking log of both the sides, we get,

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

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

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

=>

=>

=>

=>

=>

=>

### Question 12. Differentiate  with respect to x.

Solution:

We have,

=>

On taking log of both the sides, we get,

=> log y = log

=> log y = 10x log 10

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

=>

=>

=>

=>

=>

=>

### Question 13. Differentiate y = sin xx with respect to x.

Solution:

We have,

=> y = sin xx

=> sinâˆ’1 y = xx

On taking log of both the sides, we get,

=> log (sinâˆ’1 y) = log xx

=> log (sinâˆ’1 y) = x log x

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

=>

=>

=>

=>

=>

=>

=>

=>

### Question 14. Differentiate y = (sinâˆ’1x)x with respect to x.

Solution:

We have,

=> y = (sinâˆ’1x)x

On taking log of both the sides, we get,

=> log y = (sinâˆ’1x)x

=> log y = x log (sinâˆ’1x)

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

=>

=>

=>

=>

=>

### Question 15. Differentiate  with respect to x.

Solution:

We have,

=>

On taking log of both the sides, we get,

=> log y = log

=> log y = sinâˆ’1x log x

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

=>

=>

=>

=>

=>

### Question 16. Differentiate  with respect to x.

Solution:

We have,

=>

On taking log of both the sides, we get,

=> log y = log

=> log y =

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

=>

=>

=>

=>

=>

### Question 17. Differentiate  with respect to x.

Solution:

We have,

=>

On taking log of both the sides, we get,

=> log y = log

=> log y = tanâˆ’1 x log x

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

=>

=>

=>

=>

=>

### (i) y = xx âˆšx

Solution:

We have,

=> y = xx âˆšx

On taking log of both the sides, we get,

=> log y = log (xx âˆšx)

=> log y = log xx + log âˆšx

=> log y = x log x +

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

=>

=>

=>

=>

=>

### (ii)

Solution:

We have,

=>

=>

=>

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

=>

=>

=>

=>

=>

### (iii)

Solution:

We have,

=>

=>

=>

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

=>

=>

=>

=>

=>

### (iv) y = (x cos x)x + (x sin x)1/x

Solution:

We have,

=> y=(x cos x)x + (x sin x)1/x

=>

=>

=>

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

=>

=>

=>

=>

### (v)

Solution:

We have,

=>

=>

=>

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

=>

=>

=>

=>

=>

### (vi) y = esin x + (tan x)x

Solution:

We have,

=> y = esin x + (tan x)x

=>

=>

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

=>

=>

=>

### (vii) y = (cos x)x + (sin x)1/x

Solution:

We have,

=> y = (cos x)x + (sin x)1/x

=>

=>

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

=>

=>

=>

### (viii) , for x > 3

Solution:

We have,

=>

=>

=>

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

=>

=>

=>

### Question 19. Find dy/dx when y = ex + 10x + xx.

Solution:

We have,

=> y = ex + 10x + xx

=>

=>

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

=>

=>

=>

=>

### Question 20. Find dy/dx when y = xn + nx + xx + nn.

Solution:

We have,

=> y = xn + nx + xx + nn

=>

=>

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

=>

=>

=>

=>

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