Question Detail
Question
If y = e2x log(x+6), find dy/dx. (Assume log denotes natural logarithm).
Correct Answer
Option A is the correct answer.
Detailed Explanation
This problem requires the product rule: d/dx(uv) = u'v + uv'.Let u = e2x and v = log(x+6) (assuming natural log, ln(x+6)).Then u' = d/dx(e2x) = 2e2x.And v' = d/dx(ln(x+6)) = 1/(x+6).So, dy/dx = (2e2x)ln(x+6) + e2x(1/(x+6)).Factor out e2x: dy/dx = e2x (2ln(x+6) + 1/(x+6)).Rearranging the terms inside the parenthesis gives e2x (1/(x+6) + 2log(x+6)), which matches option A.
Hint
Apply the product rule for differentiation and remember that d/dx(ln(u)) = u'/u.