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Advanced Math/Calculus AB question - implicit differentiation

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Question
Hello Mr. Ahmed,

I have a multiple choice question and I am getting the derivative, but my answer choice is not listed in the problem. Hereis the problem I a struggling with:

If tan(xy)=x, then dy/dx=

A. (1-ytan(xy)sec(xy))/(xtan(xy)sec(xy))
B. (sec^2(xy)-y)/x
C. cos^2(xy)
D. (cos^2(xy))/x
E. (cos^2(xy)-y)/x


Thank you so much for your time!

Nikita

Answer
Hi Nikita,
To make it easier, let z = xy
Now,
tan(z) = x
differentiating, we get
secē(z) . dz/dx = 1
but from z = xy
dz/dx = x.dy/dx + y
Therefore,
secē(xy) . (x.dy/dx + y) = 1
x.dy/dx + y = 1/secē(xy)
x.dy/dx + y = cosē(xy)
x.dy/dx = cosē(xy) - y
dy/dx = [cosē(xy) - y] / x

Regards

Advanced Math

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