\nopagenumbers
\magnification 1900

\parskip 10pt
\parindent 0pt




Find the derivative of $(3x + 4)^{2x}$

Method 1: use implicit differentiation and logarithmic differentiation:

Write as an equation and ``simplify" until the equation is in a format in 
which you know how to take the derivative:

$y = (3x + 4)^{2x}$

$ln(y) = ln(3x + 4)^{2x}$

$ln(y) = 2x \cdot ln(3x + 4)$

Now that we have an equation where we know how to take the derivative of 
both sides, we can take the derivative 
using implicit differentiation:

${1 \over y} y' = 2 ln(3x + 4) + 2x({1 \over 3x + 4})3$

$ y' = y[2 ln(3x + 4) + {6x \over 3x + 4}] = (3x + 4)^{2x}[2 ln(3x + 4) + 
{6x \over 3x + 4}] $  


Method 2:  Use logarithmic differentiation directly:

$(3x + 4)^{2x} = e^{ln{(3x + 4)^{2x} }}$

Thus $[(3x + 4)^{2x}]' = [e^{ln{(3x + 4)^{2x} }}]'
 = [e^{2x \cdot ln{(3x + 4) }}]'$


\end
