How would I find the equation of the tangent line f(x)=3^ln(ex) at x=e

Question

I don't understand the ex part

Answer 1

This problem is meant to emphasize the chain rule when differentiating.

Recall that if y=A^x, then y’ = A^x * lnA

However, considering that the given function is composite, we must apply the chain rule, or

1)     The derivative of the outside 3 ^( ) * ln3,

2)     Evaluated at the inside or 3^(ln(ex))*ln3

3)     Times the derivative OF the inside or 3^(ln(ex))*ln3*e/ex

Therefore y’ simplifies to y’ = ln3*(3^ln(ex)) / x

Evaluate f(x) at x = e, gives y = 9 or the point (e,9) must be on both the tangent line and the function.

Then evaluating y’ at e, we get m = ln3*9/e

Using linear equation y = mx + B and the point determined at (e,9), the solution for the tangent line is:

                               Y = 9/e * ln(3) x +9 – 9*ln(3)