Question

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

Answer 1

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

Answer 2

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

Answer 3

@sweetburger what do you mean?

Answer 4

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

Answer 5

just trying to help

Answer 6

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

Answer 7

\[x \arctan(x)=e^y \\ \text{ differentiate both sides w.r.t. } x \\ \frac{d}{dx}(x \arctan(x))= \frac{d}{dx}(e^y) \\ \]


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

Answer 8

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

Answer 9

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

Answer 10

with respect to x

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

is that what you mean @meaghan25

Answer 15

derivative of that inverse function

Answer 16

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

Answer 17

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

Answer 18

yes I need the inverse

Answer 19

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

Answer 20

what do you need the inverse of ?

Answer 21

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.

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

ok thank you so much

Answer 27

do you want me to check your answer?

Answer 28

or even your work

Answer 29

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

Answer 30

alright

Answer 31

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