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

1. 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

2. sweetburger

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

3. freckles

@sweetburger what do you mean?

4. sweetburger

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

5. sweetburger

just trying to help

6. sweetburger

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

7. 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

8. 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

9. anonymous

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

10. freckles

with respect to x

11. 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

12. sweetburger

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

13. freckles

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

14. freckles

is that what you mean @meaghan25

15. freckles

derivative of that inverse function

16. freckles

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

17. freckles

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

18. anonymous

yes I need the inverse

19. freckles

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

20. freckles

what do you need the inverse of ?

21. 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.

22. freckles

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

23. freckles

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

24. freckles

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

25. freckles

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

26. anonymous

ok thank you so much

27. freckles

28. freckles

29. anonymous

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

30. freckles

alright

31. courtneygraley009

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