Formal definition of derivative: trig vs non trig functions

[LearninDaMath]

for derivative sinx = cosx, by setting up into formal definition formula limΔx->0 \frac{f(x+Δx)-f(x)}{Δx}


this formal definition of derivative is formulated from the cartesian coordinate system where the horizontal is x and vertical is y. But sinx is a trig function and trig functions are represented on the graph where the horizontal is an angle.

So how does it make sense that trig functions can be set up into the formal definition of a derivative in the same manner as non-trig functions?

Meaning, how does this make sense? limΔx->0 \frac{sin(x+Δx)-sinx}{Δx}

I know how to work it out and get cosx. But it doesn't seem to make as much sense visually as a nontrig proof.

 

[Mark44]

for derivative sinx = cosx, by setting up into formal definition formula limΔx->0 \frac{f(x+Δx)-f(x)}{Δx}


this formal definition of derivative is formulated from the cartesian coordinate system where the horizontal is x and vertical is y. But sinx is a trig function and trig functions are represented on the graph where the horizontal is an angle.

Not really. The argument to the standard trig functions is radians, which are dimensionless, which makes them just like ordinary real numbers.

So how does it make sense that trig functions can be set up into the formal definition of a derivative in the same manner as non-trig functions?

Meaning, how does this make sense? \frac{sin(x+Δx)-sinx}{Δx}

I know how to work it out and get cosx. But it doesn't seem to make as much sense visually as a nontrig proof.

 

[LearninDaMath]

Thanks Mark44, when thinking about it, it seems to be the same thing after all. The horizontal is still x in either case like you say. And since a trig function is just a function, sin(x) can be thought of as f(x). So both types of functions are essentially the same thing, such as in terms of representing two points on each respective function (trig or non-trig) in the formal definition formula. Thanks, Mark, appreciate it.