Find derivative.

[Infinity.]

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[srmichael]

Hi there,

Could I have some help to find the derivative of xtan(y)

I do know implicit differentiation will need to be used.

Please! Thanks heaps!

Please show us what you have tried so far so we know where you are stuck.

 

[Infinity.]

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[Deleted member 4993]

The full question is Let x[tan(y)] + log_e(y) = e^xy

That cannot be the full question - because it does not have a question!

Do you need to find \(\displaystyle \dfrac{dy}{dx}\) ?

If yes - then use implicit differentiation and chain rule.

tan(y) + x*sec²(y)\(\displaystyle \dfrac{dy}{dx}\) + \(\displaystyle \dfrac{1}{y}\) * \(\displaystyle \dfrac{dy}{dx}\) = e^xy * [y + x * \(\displaystyle \dfrac{dy}{dx}\)]


And now isolate \(\displaystyle \dfrac{dy}{dx}\) and solved .....

 

[srmichael]

Well tan(y) = d(tan(y)/dx = sec²(y) * dy/dx ....correct? Correct!

If that's right, what happens next?

The full question is Let x[tan(y)] + log_e(y) = e^xy

loge(y) = d(log(y))/dx = 1/y * dy/dx ...correct? Correct!

e^xy = e^xy(y + x * dy/dx) .....correct? Correct!

You have all the individual derivatives correctly done. the only thing you are mising is that for x[tan(y)] you have to use the product formula since tan(y) is a function of x.

\(\displaystyle \frac{d}{dx}x[tan(y)] = tan(y) + x[sec^2(y)\cdot\frac{dy}{dx}]\)

Now use this with the other derivatives you found to solve for \(\displaystyle \frac{dy}{dx}\)

Make sense?

 

[Infinity.]

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[srmichael]

Yeah, I need to find dy/dx

So...

tan(y) + x*sec²(y)\(\displaystyle \dfrac{dy}{dx}\) + \(\displaystyle \dfrac{1}{y}\) * \(\displaystyle \dfrac{dy}{dx}\) = e^xy * [y + x * \(\displaystyle \dfrac{dy}{dx}\)]

Therefore

tan(y) + dy/dx[xsec²(y)+ (1/y)] = e^xy * dy/dx (y+x) =====> Not quite.

Therefore

dy/dx[xsec²(y)+(1/y) / y+x] = e^xy - tan(y)

Therefore

dy/dx = e^xy - tan(y) / [xsec²(y)+(1/y) / y+x] ........correct?

If that's right, can it be simplified further?

\(\displaystyle e^{xy}(y + x\frac{dy}{dx})\) = \(\displaystyle ye^{xy} + xe^{xy}\frac{dy}{dx}\)

Now proceed....

 

[Infinity.]

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[srmichael]

tan(y) + dy/dx[xsec²(y)+ (1/y)] = ye^xy + xe^xy * dy/dx

Thus

dy/dx[xsec²(y) + (1/y) - xe^xy] = ye^xy - tan(y)

dy/dx [sec²(y)+ (1/y) - e^xy] = ye^xy - tan(y)

dy/dx = [ye^xy - tan(y)] / [sec²(y) + (1/y) - e^xy]

This right? Can it be simplified now?

Looks good! I don't believe it can be simplified any further. Someone will correct me if I am wrong.

 

[Infinity.]

Thanks dude, much appreciated!