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.)

  • 1
    $\begingroup$ This image can be helpful: physics.unsw.edu. Also check out answers to this question: math.stackexchange.com. $\endgroup$
    – jonsno
    Commented Oct 2, 2017 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. $\endgroup$ Commented Oct 2, 2017 at 9:18
  • 1
    $\begingroup$ 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
    Commented Oct 2, 2017 at 10:14

2 Answers 2


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:

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


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?


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).


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .