Implicit Differentiation: find the derivative of the function at the indicated point. lnxy+5x=30 at (1, e^25).I have figured the following so far: (1/xy)+5x(dy/dx)=0 I think that is how it goes.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find `(dy)/(dx)`  for the given function, hence, you need to differentiate with respect to x such that:

`(d(ln xy))/(dx) + (d(5x))/(dx) = (d(30))/(dx)`

`1/(xy)*(y + x*(dy)/(dx)) + 5 = 0`

You need to open the brackets such that:

`y/(xy) + (x/(xy))*(dy)/(dx) = -5`

`1/x + (1/y)*(dy)/(dx) = -5 => (1/y)*(dy)/(dx) = -5 - 1/x => (dy)/(dx) =y(-5 - 1/x)`

You need to evaluate  `(dy)/(dx)`  at `(1,e^25)`  such that:

`(dy)/(dx) =e^25(-5 - 1/1) =>(dy)/(dx) = -6*e^25`

Hence, evaluating  `(dy)/(dx)`  at the given point yields `(dy)/(dx) = -6*e^25.`

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