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While I think that ideally, even in a freshman course of calculus, students should receive some historical notions about the development of the ideas of infinitesimal calculus, I think that, even in a freshman course of calculus, the true definition of derivative of a function should be given, that is, via the first order approximation. A function $f:(a,b)\to\mathbb{R}$ is differentiable at $x$ if there exists $m$ such that $$f(x+h)=f(x)+mh+o(h)\quad \mathrm{as}\ \ h\to0.$$

The fact that the coefficient $m$ (the derivative) can be characterized, and sometimes efficiently computed, as a limit of a quotient, has certainly to be observed, and should be applied immediately to treat some elementary functions like $x^2$, $1/x$ or $e^x$, as usual. But I would never give it as a definition.

I think there is a philosophical issue here. It may seem simpler to define something as the result of a procedure for getting it, compared with defining it via a characteristic property. But the first latter way is superior, and on a long distance, simpler. And in the case of students who will stop there their mathematical education, then, I prefer they at least see the true idea behind, rather that being able to compute the derivative of $\cos(e^x)$ : when will that be of use for them?

The definition via first order expansion is very natural, and more understandable to the freshman students. It has a more direct geometrical meaning. It reflects the physical idea of linearity of small increments (like in Hooke's law of elasticity, etc). It is much closer to the practical use of derivatives in approximations. It makes easier all the elementary theorems of calculus (consider how needlessly complicated becomes the proof of the theorem for the derivative of a composition by introducing a useless quotient). Finally, it is closer to the generalization to Fréchet differential, which is a good thing for those students that will continue their study in maths.

A funny remark, from my experience. Ask students that received the definition of derivative as limit of incremental quotient, to compute $\lim_{x\to 0 }\sin(x)/x$. Will anybody say, it's the derivative of $\sin(x)$ at $0$, that is $\cos(0)=1$? No, they will try and use the "rule of de L'Hopital"!

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While I think that ideally, even in a freshman course of calculus, students should receive some historical notions about the development of the ideas of infinitesimal calculus, I think that, even in a freshman course of calculus, the true definition of derivative of a function should be given, that is, via the first order approximation. A function $f:(a,b)\to\mathbb{R}$ is differentiable at $x$ if there exists $m$ such that $$f(x+h)=f(x)+mh+o(h)\quad \mathrm{as}\ \ h\to0.$$

The fact that the coefficient $m$ (the derivative) can be characterized, and sometimes efficiently computed, as a limit of a quotient, has certainly to be observed, and should be applied immediately to treat some elementary functions like $x^2$, $1/x$ or $e^x$, as usual. But I would never give it as a definition.

I think there is a philosophical issue here. It may seem simpler to define something as the result of a procedure for getting it, compared with defining it via a characteristic property. But the first way is superior, and on a long distance, simpler. And in the case of students who will stop there their mathematical education, then, I prefer they at least see the true idea behind, rather that being able to compute the derivative of $\cos(e^x)$ : when will that be of use for them?

The definition via first order expansion is very natural, and more understandable to the freshman students. It has a more direct geometrical meaning. It reflects the physical idea of linearity of small increments (like in Hooke's law of elasticity, etc). It is much closer to the practical use of derivatives in approximations. It makes easier all the elementary theorems of calculus (consider how needlessly complicated becomes the proof of the theorem for the derivative of a composition by introducing a useless quotient). Finally, it is closer to the generalization to Fréchet differential, which is a good thing for those students that will continue their study in maths.

A funny remark, from my experience. Ask students that received the definition of derivative as limit of incremental quotient, to compute $\lim_{x\to 0 }\sin(x)/x$. Will anybody say, it's the derivative of $\sin(x)$ at $0$, that is $\cos(0)=1$? No, they will try and use the "rule of de L'Hopital"!