IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

Ashish

Hello everyone!

If y = (sinx)^cosx + (cosx)^sinx,prove that

dy/dx = (sinx)^cosx. [cotx cosx ? sinx (log sinx)] + (cosx)^sinx. [cosx (log cosx) ? sinx tanx]

(CBSE 2004C)

The answer that I got was

dy/dx = [(sinx)^cosx. (cosx)^sinx] [cotx cosx ? sinx (logsinx) + cosx (log cosx) ? sinx tanx]

Kindly explain why I am getting it different. I?ll be grateful to you.

Thanks!

With regards

Ashish

Ashish

That question mark (?) was actually meant to be the dot (.) for multiplication. I don't know what the problem is but every time it occurs. Any way out?

smartk

firstly d correct ans is

[sinx^cosx{cotxcosecx-sinxlogsinx} +cosx^sinx{cosxlogcosx+tanxsecx}]

take p= sinx^cosx then differentiate it u will get d first expression and then q= cosx^sinx . and add the outcome.

rvinitra

The Question is:
If y = (sinx)^cosx + (cosx)^sinx, prove that
dy/dx = (sinx)^cox.[cotx.cosx - sinx(log sinx)] + (cosx)^sinx.[cosx (log cosx) - sinx.tanx]

The answer is:
Consider u = (sinx)^cosx and v = (cosx)^sinx
Hence y = u + v
Differentiating with respect to x, we get
dy/dx = du/dx + dv/dx ........(1)

Now consider u = (sinx)^cosx
Taking log on both sides, we get
log(u) = log(sinx^cosx)
log(u) = cosx.log(sinx)
Differentiating with respect to x, we get
1/u . du/dx = -sinx.log(sinx) + (cosx/sinx).cosx
du/dx = u.[cotx.cosx - sinx.log(sinx)]
du/dx = (sinx)^cosx[cotx.cosx - sinx.log(sinx)].......(2)

Now consider v = (cosx)^sinx
Taking log on both sides, we get
log(v) = log(cosx^sinx)
log(u) = sinx.log(cosx)
Differentiating with respect to x, we get
1/v . dv/dx = cosx.log(cosx) + (sinx/cosx)(-sinx)
dv/dx = v.[cox.log(cosx) - tanx.sinx]
dv/dx = (cosx)^sinx[cox.log(cosx) - tanx.sinx]......(3)

Now from (1), we get dy/dx = (2) + (3)
Hence
dy/dx = (sinx)^cosx[cotx.cosx - sinx.log(sinx)] + (cosx)^sinx[cox.log(cosx) - tanx.sinx]

Hence proved.

rvinitra

I hope u got it. If not please come forward and ask any doubt u hav!!

Ashish

Hello rvinitra!

Perfect answer! Alas! I have already accepted an answer. You deserve your first salute!

Thanks!

rvinitra

thank u

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