Find the derivative of a logarithmic function

Question

Differentiate.  f(x) = log₄ xe^x

Answer 1

Let y = log₄(xe^x)

 

Convert to natural logs and simplify:    

 

y = ln(xe^x)/ln4

 

   = (1/ln4) [ln(xe^x)]

 

   = (1/ln4)[lnx + ln(e^x)]

 

   = (1/ln4)[lnx + x]

 

So, y' = (1/ln4)[(1/x )+ 1]   =  (x + 1)/(xln4)

Answer 2

d/dx (log(4) x e^x)

Factor out constants:
     = log(4) (d/dx(e^x x))
Use the product rule,

     d/dx(u v) = v ( du)/( dx) + u ( dv)/( dx),

     where u = e^x and v = x:
     = log(4)  x (d/dx(e^x)) + e^x (d/dx(x))
The derivative of e^x is e^x:
      = log(4)  (e^x (d/dx(x)) + e^x x)
The derivative of x is 1:
      = log(4) (e^x x + 1 e^x)
Simplify
      = log(4) (e^x + e^x x)