Suppose ()=ln(()) , where ()>0 for all real numbers and is differentiable for all real numbers. If (4)=2 and ′(4)=−15 , find ′(4) .

f(e) = 3 iff a*ln(2e+b) = 3 iff 2e+b = e^(3/a)

f ' (e) = 6/e iff 2a / (2e+b) = 6/e iff 2e+b = ae/3 = e/(3/a)

Then, e^(3/a) = e / (3/a)

We're looking for x such that e^x = e/x.

This kind of equation can't be solved analytically, in general,

but here we've got an obvious solution : x = 1

Hence a = 3,

and then b = -e

Step-by-step explanation:

or

step-by-step explanation:

we need to solve for 'x' the following equation:  

we have that:

⇒

⇒ ⇒

factorizing, we get that:

  ⇒

then, the solutions are: or

nothing extraneous has been found!