Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping

Differentiate it please as earlier as possible and explain it briefly

Differentiate it please as earlier as possible and explain it briefly
 

Question Image
Grade:12

1 Answers

Samyak Jain
333 Points
2 years ago
You can solve this problem directly by differentiation rules but there’s an alternative simpler method of substitution!
Substitute x = sin\theta in the given expression. Then \theta = sin– 1x. As |x| \pi/2 \theta \pi/2.
dx = cos\theta d\theta  \Rightarrow  d\theta / dx = 1/cos\theta = sec\theta          ...(1)
Given expression becomes sin\theta.\theta / \sqrt{1-sin^2\theta}  +  log\sqrt{1-sin^2\theta}   =   \theta sin\theta / |cos\theta|  +  log|cos\theta|
         =  \theta sin\theta / cos\theta  +  logcos\theta     [\because for – \pi/2 \theta \pi/2,  cos\theta > 0]
         = \theta tan\theta + logcos\theta
d(given expression) / dx  =  {d(given expression) / d\theta} {d\theta / dx}
                   = {d(\theta tan\theta + logcos\theta)/d\theta} {sec\theta}             [From (1)]
                   = [\theta sec2\theta + tan\theta .1 + {1/cos\theta}{–sin\theta}] {sec\theta}     [Apply uv rule of differentiation and formulae]
                   = [\theta sec2\theta + tan\theta – tan\theta] sec\theta
                   =  \theta sec3\theta                ...(2)
As x = sin\theta,  x2 = sin2\theta = 1 – cos2\theta.
\therefore cos2\theta = 1 – x2   i.e.   sec2\theta = 1/cos2\theta = 1/(1 – x2), sec\theta = 1/\sqrt{1-x^2}
Substitute the value of sec\theta in (1).
So, required answer is sin– 1x . (1/\sqrt{1-x^2})3  =  sin– 1x . 1/(1 – x2)3/2
                               = sin– 1x / (1 – x2)3/2 .

Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy

ASK QUESTION

Get your questions answered by the expert for free