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

### Question 1. Differentiate y = x^{1/x} with respect to x.

**Solution:**

We have,

=> y = x

^{1/x}On taking log of both the sides, we get,

=> log y = log x

^{1/x}=> log y = (1/x) (log x)

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

=>

=>

=>

=>

=>

=>

### Question 2. Differentiate y = x^{sin x} with respect to x.

**Solution:**

We have,

=> y = x

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

=> log y = log x

^{sin 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 = cos

^{−1}x log xOn 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 = e^{x log x} with respect to x.

**Solution:**

We have,

=> y=e

^{x log x}=> y =

=> y = x

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

=> log y = log x

^{x}=> 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 = 10^{log sin x} with respect to x.

**Solution:**

We have,

=> y = 10

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

=> log y = log 10

^{log 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 = 10

^{x}log 10On differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

### Question 13. Differentiate y = sin x^{x} with respect to x.

**Solution:**

We have,

=> y = sin x

^{x}=> sin

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

=> log (sin

^{−1}y) = log x^{x}=> log (sin

^{−1}y) = x log xOn differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

=>

=>

=>

### Question 14. Differentiate y = (sin^{−1}x)^{x} with respect to x.

**Solution:**

We have,

=> y = (sin

^{−1}x)^{x}On taking log of both the sides, we get,

=> log y = (sin

^{−1}x)^{x}=> log y = x log (sin

^{−1}x)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 = sin

^{−1}x log xOn 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 = tan

^{−1}x log xOn differentiating both sides with respect to x, we get,

=>

=>

=>

=>

=>

### Question 18. Differentiate the following with respect to x.

### (i) y = x^{x} √x

**Solution:**

We have,

=> y = x

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

=> log y = log (x

^{x}√x)=> log y = log x

^{x}+ log √x=> 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,

=>

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### (iii)

**Solution:**

We have,

=>

=>

=>

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

=>

=>

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=>

### (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,

=>

=>

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=>

### (vi) y = e^{sin x} + (tan x)^{x}

**Solution:**

We have,

=> y = e

^{sin 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 = e^{x} + 10^{x} + x^{x}.

**Solution:**

We have,

=> y = e

^{x}+ 10^{x}+ x^{x}=>

=>

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

=>

=>

=>

=>

### Question 20. Find dy/dx when y = x^{n} + n^{x }+ x^{x} + n^{n}.

**Solution:**

We have,

=> y = x

^{n}+ n^{x}+ x^{x}+ n^{n}=>

=>

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

=>

=>

=>

=>