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,
=>
=>
=>
=>
=>
Question 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 = e3x sin 4x 2x.
Solution:
We have
=> y = e3x sin 4x 2x.
On taking log of both the sides, we get,
=> log y = log (e3x sin 4x 2x)
=> log y = log e3x + log (sin 4x) + log 2x
=> 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 = xsin x + (sin x)x.
Solution:
We have,
=> y = xsin x + (sin x)x.
Let u = xsin x and v = (sin x)x. Therefore, y = u + v.
Now, u = xsin x
On taking log of both the sides, we get,
=> log u = log xsin 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[cosec2x – log(tanx)(cosec2x)] + (cotx)tanx[-sec2x + log(cotx)(sec2x)] =>
= (tanx)cotx[cosec2x – cosec2xlog(tanx)] + (cotx)tanx[sec2xlog(cotx) – sec2x]
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 = xcos x + (sin x)tan x
Solution:
We have,
=> y = xcos x + (sin x)tan x
=>
=>
On differentiating both sides with respect to x, we get,
=>
=>
=>
=>
(ii) y = xx + (sin x)x
Solution:
We have,
=> y = xx + (sin x)x
=>
=>
On differentiating both sides with respect to x, we get,
=>
=>
=>
=>
Question 30. Find dy/dx when y = (tan x)log x + cos2 (π/4).
Solution:
We have,
=> y = (tan x)log x + cos2 (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+ xlogx.
Solution:
We have,
=> y = (log x)x+ xlogx
Let u = (log x)x and v = xlogx. 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 = xlogx
On taking log of both the sides, we get,
=> log v = log xlogx
=> 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 x13y7 = (x+y)20, prove that
Solution:
We have,
=> x13y7 = (x+y)20
On taking log of both the sides, we get,
=> log x13y7 = log (x+y)20
=> log x13 + log y7 = 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 x16y9 = (x2 + y)17, prove that
Solution:
We have,
=> x16y9 = (x2 + y)17
On taking log of both the sides, we get,
=> log x16y9 = log (x2 + y)17
=> log x16 + log y9 = log (x2 +y)17
=> 16 log x + 9 log y = 17 log (x2 + y)
On differentiating both sides with respect to x, we get,
=>
=>
=>
=>
=>
=>
=>
=>
=>
Hence proved.
Question 35. If y = sin xx, prove that
Solution:
We have,
=> y = sin xx
Let u = xx. Now y = sin u.
On taking log of both the sides, we get,
=> log u = log xx
=> log u = x log x
On differentiating both sides with respect to x, we get,
=>
=>
=>
=>
Now, y = sin u
=>
=>
=>
Hence proved.
Question 36. If xx + yx = 1, prove that
Solution:
We have,
=> xx + yx = 1
=>
=>
On differentiating both sides with respect to x, we get,
=>
=>
=>
=>
=>
=>
Hence proved.
Question 37. If xy × yx = 1, prove that
Solution:
We have,
=> xy × yx = 1
On taking log of both the sides, we get,
=> log (xy × yx) = log 1
=> log xy + log yx = 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 xy + yx = (x+y)x+y, find dy/dx.
Solution:
We have,
=> xy + yx = (x+y)x+y
=>
=>
On differentiating both sides with respect to x, we get,
=>
=>
=>
=>
=>
Question 39. If xm yn = 1, prove that
Solution:
We have,
=> xm yn = 1
On taking log of both the sides, we get,
=> log (xm yn)= log 1
=> log xm + log yn = 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 yx = ey−x, prove that
Solution:
We have,
=> yx = ey−x
On taking log of both the sides, we get,
=> log yx = log ey−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.