How to use logarithmic differentiation to find the derivative of a function, and can it be used for any function?
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"Degree to radians. Express in terms of pi.?"
Yes. It can be used for any function. However, it's usually not necessary.
In its general form, it produces an "obvious" result that makes it seem like it's a waste of time:
If #y=f(x)#, then #ln(y)=ln(f(x))#. Now differentiate both sides with respect to #x# to get, by the Chain Rule, #\frac{1}{y}\frac{dy}{dx}=\frac{1}{f(x)}f'(x)#. Now solve for #dy/dx# to get #dy/dx=y\cdot \frac{f'(x)}{f(x)}=f(x)\cdot \frac{f'(x)}{f(x)}=f'(x)#.
However, for certain kinds of examples, it's not a waste of time because it gives the answer when no other approach seems to work. This is especially true in situations where the property #ln(a^{b})=bln(a)# is helpful in reducing an "atypical" function to a product of two "typical" functions.
For example, if #y=f(x)=x^{x}#, then logarithmic differentiation must be used (this function is neither a power function nor an exponential function):
#y=x^{x}\Rightarrow ln(y)=ln(x^{x})=x\ln(x)\Rightarrow #
#\frac{1}{y}\frac{dy}{dx}=ln(x)+x\cdot\frac{1}{x}=ln(x)+1#
Hence, #\frac{dy}{dx}=y(ln(x)+1)=x^{x}(ln(x)+1)#
