Differentiation Tricks

Question

Since most derivatives are trivial to take, it's understandable why integrals get most of the mathematical tricksters' attention. However, not all derivatives are trivial to take and I think it's good to have as many tricks up your sleeves as possible. I noticed the other day that $\left(\frac{dy}{dx}\right)^{-1}=\frac{dx}{dy}$ and that we can use this fact to differentiate functions like the inverse trig functions e.g. $y=\arcsin(x)$, $$\frac{d \sin(y)}{dy}=\cos(y)=\cos(\arcsin(x))=\sqrt{1-x^2}=\left(\frac{dy}{dx}\right)^{-1}$$ My question is, does anybody know of any similar tricks to differentiate those rare functions which are non-trivial?

Answer 1

$$\frac{d(u^v)}{dx} = \frac{d(e^{v \log u})}{dx} = e^{v \log u}\frac{d(v \log u)}{dx} = u^v (\frac{dv}{dx}\log u + v\frac{du/dx}{u})$$

$$\frac{d(\log_u v)}{dx} = \frac{d}{dx}\left( \frac{\log v}{ \log u} \right)$$

$$\arccos x = \frac{\pi}{2} - \arcsin x \therefore \frac{d}{dx}\arccos x = - \frac{d}{dx}\arcsin x \text{, and so on with $\arctan$ and arccotan.}$$

Answer 2

Logarithmic differentiation.

Given $y=f(x)>0$, we can take $\ln y = \ln f(x)$, and find that $\dfrac{dy}{dx} = f(x) \dfrac{d}{dx} \ln f(x)$.

However, if $|y|=|f(x)|$, then we obtain $\ln |y| = \ln |f(x)|$, and then $\dfrac{d}{dx} \ln |y| = \dfrac{y}{|y|^2}y' =\dfrac{y'}{y}.$

So in fact whether $f(x)$ is positive or negative is of no concern.

This can be used on rational functions, to avoid using the quotient rule, or on functions of the form $y=(f(x))^{g(x)}$.