Question

How to use logarithmic differentiation to find the derivative of a function, and can it be used for any function?

Answer 1

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)`