Question Detail
Question
If y = e2x log(x+6), find dy/dx. (Assume 'log' denotes the natural logarithm, 'ln')
Correct Answer
Option A is the correct answer.
Detailed Explanation
We need to differentiate y = e2x ln(x+6) using the product rule: (uv)' = u'v + uv'.Let u = e2x, then u' = d/dx(e2x) = 2e2x (using the chain rule).Let v = ln(x+6), then v' = d/dx(ln(x+6)) = 1/(x+6) (using the chain rule).Now apply the product rule:dy/dx = (2e2x)ln(x+6) + e2x(1/(x+6))Factor out e2x:dy/dx = e2x [2ln(x+6) + 1/(x+6)]. This matches option A.
Hint
Remember the product rule for differentiation and the chain rule for exponential and logarithmic functions.