# Problem in evaluating logarithm derivatives

Given the property of the logarithm that $$\log{xy} = \log{x} + \log{y}$$, how would one take the 'derivative' of this?

To be more clear,

$$\log{xy} = \log{x} + \log{y}$$ (property of $$\log$$)

$$D(\log{xy}) = D(\log{x} + \log{y})$$ (i) (Take derivative on both sides)

Now, $$D(\log{x} + \log{y}) = D(\log{x}) + D(\log{y})$$ (ii) (Derivative of sum is sum of derivatives)

Combining (i) and (ii): $$D(\log{xy}) = D(\log{x}) + D(\log{y})$$ (iii)

Implies: $$\frac{1}{xy} = \frac{1}{x} + \frac{1}{y}$$ (Evaluate derivative of logarithm using $$D(\log{x}) = \frac{1}{x}$$

$$\frac{1}{xy} = \frac{1}{x} + \frac{1}{y}$$ looks false to me; e.g. while $$\log{6}$$ does equal $$\log{2} + \log{3}$$, $$\frac{1}{6}$$ does not equal $$\frac{1}{2} + \frac{1}{3}$$.

My first guess was that the issue was related to what variable I take the derivative with respect to, but I'd like to understand this a little more formally if someone could guide me.

What's going wrong in this example?

• Like the comment above said- you can’t take derivatives with respect to all different variables and expect them to be equal. Jan 5, 2019 at 4:19
• (This comment was from me -- deleted when reading the fact that ayt the end of your question you acknowledged something was sketchy in the variables you differentiated with respect to. Essentially, I was saying you were "differentiating simultaneously wrt 3 variables," in different places: $xy$, $x$, $y$ -- and that's not allowed.) Jan 5, 2019 at 4:21
• Thanks guys; this clears things up! I appreciate the fast responses. Jan 5, 2019 at 4:24

Yes, as you noted, it matters which variable you take a derivative with respect to. Since we have two variables, if we assume they are independent, then we need to use partial derivatives. If we chose $$x$$, then $$\frac{\partial \log(xy)}{\partial x}=\frac{y}{xy}=\frac1x$$and $$\frac{\partial}{\partial x}(\log x+\log y)=\frac1x+0=\frac1x$$So the answers agree. (the same thing happens if we chose $$y$$ instead).
Alternatively, you could take a total derivative. $$\mathrm d(\log xy)=\frac{\partial \log xy}{\partial x}\mathrm dx+\frac{\partial \log xy}{\partial y}\mathrm dy=\frac1x\mathrm dx+\frac1y \mathrm dy$$This agrees with $$d(\log x+\log y)=\frac1x\mathrm dx+\frac1y \mathrm dy$$ So there are no inconsistencies.
The issue is that your differential operator $$D$$ does not behave in the way you think it does! $$D(\log(xy))=\frac{1}{xy}D(xy)=\frac{1}{xy}(ydx+xdy)=\frac{1}{x}dx+\frac{1}{y}dy=D(\log(x))+D(\log(y))$$ At no point in this computation do we actually have $$\frac{1}{xy}=\frac{1}{x}+\frac{1}{y}$$ -- we are working with differentials, and not derivatives.