I'm doing calculus and I want to be an engineer so I would like to understand the essence of the logic of implicit differentials rather than just memorizing the algorithm. Yes, I could probably memorize it and get a 100% on a test, but it means nothing unless I understand it and can acquire a practical understanding of it. I would really appreciate if someone can enlighten me.
What I understand already:
I understand how the derivatives of normal functions are found the long way… i.e.
$ f'(x)=\frac{f(x+h)-f(x))}{(x+h)-x} $
I understand that the derivative of the above equation is found when we find the limit as h, representing the distance between the two points, approaches 0.
I also understand that the chain rule, quotient rule, etc. are just algorithms that speed up the process of finding the derivative. I don't need a proof for that
An example problem
When we do implicit differential equations such as this one:
A ladder is 8.5 m long leaning against a wall, the bottom part of the ladder is 6.0 m from the wall and is sliding away from the wall at a rate of 2.5m/s.
$x^2 + y^2 = h^2$ (Pythagorean theorem) (x is x value, y is y value, h is hypot)
We can find that y = 6.2m.
The derivative is
$2x\frac{\mathrm{d} x}{\mathrm{d} t} + 2y\frac{\mathrm{d} y}{\mathrm{d} t} = 2h\frac{\mathrm{d} h}{\mathrm{d} t}$
What I don't understand: (although I can do them by memorizing the algorithm)
Is the implicit differential… How do we relate all the terms to the change in time? I mean how do we know we can make that equation's derivative with respect to t? Would it be possible to make them all with respect to the change in x? If so please demonstrate, I think that would help a lot as my biggest lack of understanding is how to know what the bottom term should be for each derivative term.
I really appreciate any help, thanks.
Best Answer
Don't treat implicit differentiation as an idea that is distinct from "regular" differentiation. Just as the chain rule is involved in every derivative, and is not a separate rule that governs only certain situations, e.g. $$y=x^2=(x)^2$$ $$ \begin{align} \frac{d}{dx}(y)&=\frac{d}{dx}(x)^2\\ &=2(x)^1\frac{d}{dx}(x)\\ &=2x\cdot 1\\ &=2x \end{align} $$ implicit differentiation is something that you're doing all the time, you just don't see it as such. For example, when you differentiate $y=3x^2+2$, view it as differentiating both sides with respect to the symbol $x$: $$ \begin{align} \frac{d}{dx}(y)&=\frac{d}{dx}(3x^2+2)\\ &=\frac{d}{dx}(3x^2)+\frac{d}{dx}(2)\\ &=3\frac{d}{dx}(x^2)+0\\ &=3\cdot 2(x)^1\frac{d}{dx}(x)\quad\text{ (via chain rule)}\\ &=6\cdot x\cdot 1\\ &=6x \end{align} $$ I think that there are two things that aid in understanding, both of which treat differentiation in a symbolic manner.
The only other thing to remember is that the chain rule needs to be applied to all symbols as in: $$ \frac{d}{d\square}(y)^2=2(y)^1\frac{d}{d\square}(y) $$