Expert: Ahmed Salami - 3/22/2009

Question
Why is the derivative of e^((1/2)x) 1/2e^((1/2)x)? I thought the rule was f'(e^kx)=(e^kx)/k. Shouldn't the answer be 2e^((1/2)x)? Thank you.

Answer
Hi Mel,
f'(e^kx) = k(e^kx)
Lets rewrite in a simpler form and consider it as a composite function.
y = e^kx
and let u = kx
we now have,
y = e^u
dy/du = e^u
and
du/dx = k
but,
dy/dx = (dy/du).(du/dx)
     = e^u . k
     = k(e^u)
     = k(e^kx)

Hope it helps.

Regards