• We will examine the derivatives of the sine and cosine functions both graphically and using the formal definition of the derivative.

Recall the graph of the function \(f(x)=\sin(x)\), where \(x\) is in radians.

At this point, we are familiar with how to sketch the graph of the first derivative, \(f'(x)\), of a function, given a graph of the original function \(f(x)\).

Starting with a sketch of the function \(f(x)=\sin(x)\), take some time now and try to produce a rough sketch of the graph of the derivative.

We can further explore the derivative of \(\sin(x)\) using the following Maple investigation.

In this investigation, you will see the graph of \(f(x) = \sin(x)\) and a tangent line drawn at one point on the left side of the graph.

If you use the slider to slide the point on the graph to the right, you will see a numerical computation of the slopes of tangent lines to the function \(y=\sin(x)\) at numerous points as the tangent travels from the left side to the right side of the graph.

Simultaneously, you will also see a graph appear that is plotting the numerical values \((x, f'(x))\) on the same set of axes.

If we were to draw a continuous curve through these plotted points, what function would you see?

How close is this sketch of the derivative to your own?