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Recently I showed my students how to prove that the derivative of $\sin(x) = \cos(x)$, using the limit definition of the derivative, trigonometric identities, and the fact that $\lim_{h \to 0} \frac{sin(h)}{h} = 1$.

I'm trying to think of another way for them to think about the derivative of $\sin(x)$. Can I say something about the periodicity - the derivative is a function that is still $2\pi$-periodic? That doesn't sound enlightening. Can I say that the derivative changes the evenness / oddness of the function, e.g. the derivative of sin(x), an odd function, is cos(x), an even function? That sounds a little more interesting but still not something that would grab their attention.

Is there a deeper way of thinking about the derivative of $\sin(x)$?

(This is for a first term course in Calculus, so no power series, please.)

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    $\begingroup$ This image can be helpful: physics.unsw.edu. Also check out answers to this question: math.stackexchange.com. $\endgroup$ – jonsno Oct 2 '17 at 9:17
  • $\begingroup$ I don't think it gets much "deeper" than deriving it from the definition. But, having the students make a rough sketch of the derivative of $\sin$, by approximating slopes of tangent lines and plotting should be edifying. $\endgroup$ – David Mitra Oct 2 '17 at 9:18
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    $\begingroup$ What about complex numbers ? Consider the function $f$ defined on $\mathbb{R}$ by : $\forall t \in \mathbb{R}, \; f(t) = \exp(it)$. One could say that $f$ describes the position of a point moving on the unit circle $\lbrace z \in \mathbb{C}, \; \vert z \vert = 1 \rbrace$. The derivative of $f$ is given by : $\forall t \in \mathbb{R}, \; f'(t) = i\exp(it) = i\cos(t) - \sin(t)$. By identification, one gets that $\sin' = \cos$. $\endgroup$ – pitchounet Oct 2 '17 at 10:14
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Apropos "deeper way":

1) $f(x) = f(-x),$ even fct.

Examples: $y=x^2$, $y=cos(x)$

$ f'(x) = -f'(-x),$ chain rule, odd fct.

2) $f(x)=-f(-x)$, odd fct.

Examples: $y=x^3,$ $ y=sin(x)$.

$f'(x) = f'(-x)$, chain rule, even fct.

3) Example, periodic fct:

$f(x) = f(x +2πk)$, $k \in \mathbb{N}$.

$f'(x)=f'(x+2πk)$.

4) Draw $\sin$ and $\cos$ curve, $0 \le x\le π/2$,

(in one diagram, superimpose).

Choose any $x_0$ in this interval.

Find the derivative of the $\sin$ fct at $x_0$ by inspection. ($\cos(x_0)$ on $\cos$ curve).

5) By inspection:

Find the derivative of $\sin(x)$ at the point of intersection of the 2 curves.

Given that at the point of intersection, $x_0=π/4$, $\sin(x_0) =(1/2)√2$, find the derivative of $\sin$ at this point.

Helps a little?

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I'd try a "dynamical" approach, using harmonic oscillators as an example. You can explain harmonic motion projecting circular motion on an axis, and referring to cosine and sine as position and speed (hence, deriving).

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