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I know that in problems dealing with rates of change that implicit differentiation is used, however how does this compare to just "regular" differentiation? The terminology differentiation just means finding the derivative

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  • $\begingroup$ The counterpart to implicit differentiation is explicit differentiation (albeit the latter term is rarely used except for emphasizing this distinction). These names follow naturally from the concept of implicit and explicit functions. $\endgroup$ – Corellian Dec 11 '15 at 1:58
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Implicit differentiation, if you ask me, is slighly confusingly named. The "implicit" does not refer to the act of differentiation, but to the function being differentiated.

Implicit differentiation means "differentiating an implicitly defined function".

For example, take the equation $$x^2+y^2=1$$

This is a simple equation of two variables, but you can understand it another way. Around each point on the unit sphere (except at $x=\pm 1$), this equation implicitly defines a function $y=y(x)$ which satisfies the equation $x^2 + y(x)^2 = 1$ for all $x$ near that point.

Now, what if we want to calculate the derivative of $y$?

Well, we need to use implicit differentiation. Differentiating the equation gives us $$2x + 2y(x)y'(x)=0$$

and notice that $y'$ crept up in there because $y$ is a function.

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  • $\begingroup$ I agree with your characterization, but I'm curious: Is this explanation of why it's called "implicit differentiation" historically accurate? (I'm not implying it isn't, mind you, I'm genuinely curious.) $\endgroup$ – Brian Tung Dec 11 '15 at 2:31
  • $\begingroup$ We do not actually "differentiate the equation". (Beginners go nuts when we say things like this because they have no idea what it means.) Suppose $x^2+y^2-1=0$. We can differentiate the constant $0$ with respect to $x$. And $x^2+y^2-1$ is just $0$ in a fancy dress. $\endgroup$ – DanielWainfleet Dec 11 '15 at 4:01
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With "regular" differentiation (bad term, I know) you can write one variable as a function of the other and then just go to town taking the derivatives. But what happens when you cannot express one as a function of the other? That is where the technique of implicit differentiation comes in. It does just get you the derivative, but for expressions that are difficult to solve "regularly".

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