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

• Last Updated : 26 May, 2021

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 = 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 = 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

=> sin−1 y = xx

On taking log of both the sides, we get,

=> log (sin−1 y) = log xx

=> log (sin−1 y) = x log x

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

=>

=>

=>

=>

=>

=>

=>

=>

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,

=>

=>

=>

=>

=>

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,

=>

=>

=>

=>

=>

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 x

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

=>

=>

=>

=>

=>

(i) y = xx √x

Solution:

We have,

=> y = xx √x

On taking log of both the sides, we get,

=> log y = log (xx √x)

=> log y = log xx + 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,

=>

=>

=>

=>

=>

(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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