00:01
We want to find the derivative evaluated at 1 for the function f of x equals 2 times x raised to e to the x.
00:10
So for this one, we first want to find the general form of the derivative.
00:16
And to do that, we had to apply natural log first since our exponent or power for this function is another function.
00:26
So applying natural log both sides we have ln of f of x.
00:33
This should equal to natural log of 2 times x raised to e to the x, which will then simplify into e to the x, ln of 2x.
00:46
So now that we have rewritten this in terms of natural log, we will then differentiate both sides.
00:55
So we have 1 over f of x times f prime of x, that's equal to, by the product rule, that's e to the x times the derivative of ln of 2x.
01:11
That'll be 1 over 2x times 2.
01:15
Plus we have ln of 2x times the derivative of e to the x, which is itself, e to the x...