Expert: Scotto - 6/22/2008

Question
I have finished most of my calculus homework assignment, but I can't seem
to figure out these problems.  Hopefully you can help me out.

1) find the derivative of the functions
   A) f(x) = ln(e^x^2+1)
   B) f(x) = x^2e^x+e^x
2) using logarithmic differentiation to find the derivative of:
  f(x) = (2x^3+1) (x^2+2)^3
3)find an equation of the tangent line to the graph of:
  f(x) = x^2+2 ln x  at it's inflection point.  (Hint: f"(x))
4) find the absolute extrema of the function:
 g(x) = (2x-1)e^-x on [0,infinity]
5) find the second derivative of:
  y = e^3x ln 2x

6) solve for t:
  5e^-2t = 6  

Answer
The derivative of ln(f(x)) is f'(x)/f(x).

The derivative of e^f(x) is f'(x)e^f(x).

The derivative is the slope at that point and a line can be found in the point-slope form using that x as x0 and f(x) as y0.

2) I'm not sure what logarithmic differentiation is.

3) Take the derivative and set it equal to 0 to find extreme points.  The derivative of ln(x) is just 1/x.  Take the second derivative and set it equal to 0 to find any inflection points.

4) Put in 0, take the limit as x goes to infinity, and find where the derivative is 0.  Note that the derivative is like a product rule.  If f(x)=g(x)^h(x), the derivative is g'(x)^h(x) + h'(x)(g(x)^h(x)).

5) This is a product rule.  The derivative of e^(ax) is ae^(ax) and the derivative of ln(ax) is a/x.  Remember the product rule:
(f(x)g(x))'=f(x)g'(x) + g(x)f'(x).

6) To both sides of the equation, do the following:
Divide by 5, take the natural ln, then divide by -2.