find the derivative of the function: x arctan x=e^y at the point (1, ln pi/4)

- anonymous

find the derivative of the function: x arctan x=e^y at the point (1, ln pi/4)

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- freckles

to differentiate x*arctan(x) (w.r.t x)
you nee to apply product rule
to differentiate e^y (w.r.t x) use chain rule

- sweetburger

you could make x=e^y ln(x)=y

- freckles

@sweetburger what do you mean?

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## More answers

- sweetburger

converting exponential to logarthimic they are equivalent... unless I am wrong idk

- sweetburger

just trying to help

- sweetburger

a^y=x equal loga(x)=y

- freckles

\[x \arctan(x)=e^y \\ \text{ differentiate both sides w.r.t. } x \\ \frac{d}{dx}(x \arctan(x))= \frac{d}{dx}(e^y) \\ \]
oh are you saying she can write:
\[\ln(x \arctan(x))=y \text{ instead of } x \arctan(x)=e^y\]
ok I was just confused because we don't have the equation x=e^y

- sweetburger

well my bad I completely read teh question wrong I thought it was 2 different equations that we were taking the derivative of... my bad

- anonymous

I don't understand what (w.r.t.x) means

- freckles

with respect to x

- anonymous

ohh ok so find the derivative of x arctanx x and then e^y the apply the product rule and then the chain rule I am looking for the inverse tho

- sweetburger

so wrtx is basically differentiating both sides with respect to x or (d/dx) of both sides?

- freckles

\[\frac{d}{dx} \arctan(x)=\frac{1}{1+x^2}\]

- freckles

is that what you mean @meaghan25

- freckles

derivative of that inverse function

- freckles

\[\frac{d}{dx}e^x=e^x \\ \frac{d}{dx}e^u=\frac{du}{dx} e^u \text{ where} u=u(x)\]

- freckles

now all you have to really do is replace u with y
and apply that product rule I was talking about

- anonymous

yes I need the inverse

- freckles

yes d/dx means differentiating w.r.t x

- freckles

what do you need the inverse of ?

- anonymous

sorry I hate open study cus of the typing issue I have with it but the directions say finding dy/dx at a point. in exercises 19 -22, find dy/dx at the given point for the equation
21. x arctan x = e^y , (1, lnpi/4)
does that mean I need to find the inverse of the function or just the derivative of dy/dx.

- freckles

the question says nothing about finding the inverse of anything
\[x \arctan(x)=e^y \\ \text{ differentiate both sides w.r.t. x} \\ \frac{d}{dx} x \arctan(x)=\frac{d}{dx}e^y \\ \text{ apply product rule on left hand side } \\ x \frac{d}{dx} \arctan(x)+\arctan(x) \frac{d}{dx} x=\frac{d}{dx}e^y \\ \text {apply chain rule to right hand side } \\ x \frac{d}{dx} \arctan(x)+\arctan(x)\frac{d}{dx}x=\frac{dy}{dx} e^y\]

- freckles

dy/dx means find the derivative of the y w.r.t. x

- freckles

you need to replace d/dx arctan(x)
and replace d/dx x
with arctan(x)'s derivative and x's derivative respectively

- freckles

anyways once you are done enter in your point and solve for dy/dx
and you are done

- anonymous

ok thank you so much

- freckles

do you want me to check your answer?

- freckles

or even your work

- anonymous

no I have the answer already just couldn't figure out what I needed to do exactly

- freckles

alright

- courtneygraley009

Try this online calculator to solve the problem http://www.acalculator.com/quadratic-equation-calculator-formula-solver.html I hope it is helpful.

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