Is this notation good for the chain rule derivative?

Question

When we take this derivative, for example:

$$y = \log(\sin x)$$ We call $u = \sin x$, so we have:

$$\frac{dy}{dx} = \frac{d y}{du}\frac{du}{dx} = \frac{1}{u}\cos x = \frac{\cos x}{\sin x}$$

But for me, it's better to do:

$$\frac{d\log\color{Blue}{\sin x}}{d\color{Blue}{\sin x}}\frac{d\sin \color{Red}{x}}{d\color{Red}{x}} = \frac{1}{\color{Blue}{\sin x}}\cos \color{Red}{x}$$ It makes easy to do the 'pattern-matching' just by looking at the differentials. No substitution. I know that $\frac{d \log[\mbox{something}]}{d[\mbox{something}]} = \frac{1}{\mbox{something}}$ for example.

However, it looks 'hairy' when I try with larger derivatives, like, for the function:

$$(13x^2-5x+8)^{\frac{1}{2}}$$ we do: $$\frac{d(13x^2-5x+8)^{\frac{1}{2}}}{dx} = \frac{d\color{Green}{(13x^2-5x+8)}^{\frac{1}{2}}}{d\color{Green}{(13x^2-5x+8)}}\frac{d(13x^2-5x+8)}{dx} = \frac{1}{2\sqrt{\color{Green}{13x^2-5x+8}}}(26x -5)$$ but it's really better for me to do like this, instead of doing the bla bla bla of changing variables and stuff. But I'm afraid my teacher does not accept this. Is this notation/way of doing good for you guys?

One more example: $$\frac{d}{dx}\sqrt{(\sin(7x+\ln(5x)))} = $$

$$\frac{d[\color{Blue}{\sin(7x+\ln(5x))}]^{1/2}}{d[\color{Blue}{\sin(7x+\ln(5x))}]}\frac{d[\sin\color{Red}{(7x+\ln(5x))}]}{d[\color{Red}{7x+\ln(5x)}]}\left[\frac{d[7\color{Purple}{x}]}{d[\color{Purple}{x}]} + \frac{d[\ln(\color{Purple}{5x})]}{d[\color{Purple}{5x}]}\frac{d[5x]}{d[x]}\right] = $$ $$\frac{1}{2}\left[\color{Blue}{\sin(7x+\ln(5x))}\right]^{-1/2}\cdot\cos(\color{Red}{7x+\ln(5x)})\left[7 + \frac{1}{\color{Purple}{5x}}\cdot 5\right]$$ So we get rid of the substitution!

(づ｡◕‿‿◕｡)づ $\ \ u, v, y$ go away!

Answer 1

This notation is absolutely acceptable, more than fine, and often used extensively in integral calculus, especially by professional mathematicians.

In this context, it is very convenient because the integration can be seen in all aspects. In other words, you are integrating with respect to a complicated expression and the form of the integrand makes sense with this expression and you want to show that clearly. In essence, it is sometimes used to be more explicit. It expresses steps more clearly at times and has the benefit of being able to visualize equalities more easily.

Be warned that this notation does get overcomplicated at times and you will want to add some substitutions for sanity, but for simpler calculations, it can be useful and looked upon favorably. You should look to study differential calculus as a topic in its own right, where this notation is used as well (in summary, the "denominator" of the derivative $\text{d}[\text{something such as} \,x]$ is eliminated and we just deal with what are called differentials). Your instructor is very intolerant, indeed, if he doesn't allow this.

As a matter of fact, there was a point towards the end of my high school career (at this point, I was studying beyond what I was enrolled in) I used this notation in a "Calculus II" type class (class focusing on expressing calculus with infinite series and basically a structured introduction to approximating evaluations of functions using different types of series), and my teacher (actually a PhD, but taught calculus at my high school in his spare time - shoutout to Dr. Brandell), and my mentor in my early mathematics self-study, commended me and demonstrated this point of view to the rest of my peers. This is very innovative actually for a young mathematician and shows promise. I'd even be willing to talk to to your instructor personally and make an argument in your defense, backed by some of my former and current professors.

Good luck in your pursuit of knowledge in mathematics and keep discovering ways to improve your notation (and, most importantly, keep asking questions)!