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anonymous

  • one year ago

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

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  1. freckles
    • one year ago
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    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
    • one year ago
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    you could make x=e^y ln(x)=y

  3. freckles
    • one year ago
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    @sweetburger what do you mean?

  4. sweetburger
    • one year ago
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    converting exponential to logarthimic they are equivalent... unless I am wrong idk

  5. sweetburger
    • one year ago
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    just trying to help

  6. sweetburger
    • one year ago
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    a^y=x equal loga(x)=y

  7. freckles
    • one year ago
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    \[x \arctan(x)=e^y \\ \text{ differentiate both sides w.r.t. } x \\ \frac{d}{dx}(x \arctan(x))= \frac{d}{dx}(e^y) \\ \] 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
    • one year ago
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    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
    • one year ago
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    I don't understand what (w.r.t.x) means

  10. freckles
    • one year ago
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    with respect to x

  11. anonymous
    • one year ago
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    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
    • one year ago
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    so wrtx is basically differentiating both sides with respect to x or (d/dx) of both sides?

  13. freckles
    • one year ago
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    \[\frac{d}{dx} \arctan(x)=\frac{1}{1+x^2}\]

  14. freckles
    • one year ago
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    is that what you mean @meaghan25

  15. freckles
    • one year ago
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    derivative of that inverse function

  16. freckles
    • one year ago
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    \[\frac{d}{dx}e^x=e^x \\ \frac{d}{dx}e^u=\frac{du}{dx} e^u \text{ where} u=u(x)\]

  17. freckles
    • one year ago
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    now all you have to really do is replace u with y and apply that product rule I was talking about

  18. anonymous
    • one year ago
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    yes I need the inverse

  19. freckles
    • one year ago
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    yes d/dx means differentiating w.r.t x

  20. freckles
    • one year ago
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    what do you need the inverse of ?

  21. anonymous
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    dy/dx means find the derivative of the y w.r.t. x

  24. freckles
    • one year ago
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    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
    • one year ago
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    anyways once you are done enter in your point and solve for dy/dx and you are done

  26. anonymous
    • one year ago
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    ok thank you so much

  27. freckles
    • one year ago
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    do you want me to check your answer?

  28. freckles
    • one year ago
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    or even your work

  29. anonymous
    • one year ago
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    no I have the answer already just couldn't figure out what I needed to do exactly

  30. freckles
    • one year ago
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    alright

  31. courtneygraley009
    • one year ago
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    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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