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 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 = 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,

=> 

=> 

=> 

=> 

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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,

=> 

=> 

=> 

=> 

=> 

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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,

=> 

=> 

=> 

=> 

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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 10

On 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 x

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

=> 

=> 

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

Solution:

We have, 

=> y = (sin⁻¹x)^x

On taking log of both the sides, we get,

=> log y = (sin⁻¹x)^x

=> log y = x log (sin⁻¹x)

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

=> 

=> 

=> 

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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⁻¹x log x

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

=> 

=> 

=> 

=> 

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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,

=> 

=> 

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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 x

On 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 √x

On 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,

=> 

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(ii) 

Solution:

We have,

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

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(iii) 

Solution:

We have,

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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,

=> 

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(v) 

Solution:

We have,

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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,

=> 

=> 

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(viii) , for x > 3

Solution:

We have,

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

=> 

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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 = xⁿ + n^x + x^x + nⁿ.

Solution:

We have, 

=> y = xⁿ + n^x + x^x + nⁿ

=> 

=> 

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

=> 

=> 

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=>