Question

derivative of sin(x) does not correlate with the positive and negative slopes of sin(x)

I know that the derivative of sin(x) is cos(x).

But, on a graph the derivative does not match up with when the slope of sin is positive, and when it is negative.

why is this? or am i looking at something wrong?

Answer 1

if you observe both sin and cos together, you will notice that
cos (which is the value of derivative of sin) is always positive as long as sin is increasing like in the range [0,90], but once the value of sin starts decreasing i.e, once it's slope becomes negative, it is reflected as negative value of cos function at the exact time instants

Answer 2

A positive slope means a line moves up from left to right.
A negative slope means the lines slants down from left to right.

Here is an annotated graph. It should be clear that the zero slopes of the sin i.e. at the max and min points) occur at the same x-values where cos(x) = 0.

Answer 3

The short answer is that if you aren't seeing the correlation then you *are* looking at it wrong, which isn't a criticism because it takes a while to develop this intuition. Working with these two functions is a way to develop that intuition. Start at 0: cos(0) = 1, so that means the slope of the sine graph should be 1 at x = 0. Whether it looks that way or not depends on whether the graph is drawn to appropriate scale (often it is not). Then look at the places where the sine graph has max or min values. It's level at those points, so that means the cosine graph should be crossing the x-axis at those points (cosine = 0 when the slope of sine = 0). I urge you to spend some time studying these graphs and understanding their relationship because the effort will pay off in terms of improved intuition both with respect to these functions and with respect to derivatives in general.