Let's take a look at another family of functions, the trigonometric ones. Let's start with the sine function. Let's consider a few particular points. Minus Pi over 2, Pi over 2, 0, and minus Pi. Let's look at the slopes at this point. The slope at Pi over 2 is 0, the slope at minus Pi over 2 is 0, the slope at 0 is actually 1, and the slope at minus Pi, it's going to be minus 1, that's the one on the left. Now let's compare it with the function cosine. Now we're not going to look at the slopes, we're actually going to look at the values. At Pi over 2, the value of cos is 0, at minus Pi over 2 is 0. At 0 it's 1 and at minus Pi it's minus 1. Do you think that's a coincidence? Actually, it's not. It's always the case that if f of x is sine of x, then the derivative f prime of x is the cosine of x. Why would this happen? We'll take a look at it in a minute, but it's a really, really nice property. Now, the same thing happens in the opposite way, except with the minus sign. If this is cosine, let's take a look at some slopes. At 0 minus Pi, Pi over 2 and minus Pi over 2. At 0, the slope is 0, at minus Pi the slope is 0 again. At Pi over 2 it's minus 1, and at minus Pi over 2 it is 1 Now let's look at sine of x, but the values now. At 0 you get 0, at minus Pi you get 0. At Pi over 2 you get 1 and at minus Pi over 2 you get a minus 1. It's almost the same, we just have to flip it with a sign. If f of x is cos x, then derivative is negative sine of x. Again, we'll see why in a minute, but this is something to remember. The derivative of sine is cosine and the derivative of cosine is negative sine. There's a nice geometric reason why this works and take a look. Let's take a circle of radius 1 and let's draw any radial line and x is the angle between the horizontal axis and the line. The projection over the x-axis is cosine effects and the projection over the y-axis is sine of x. Now let's move the angle x slightly by a quantity of Delta x. Now the projection on the horizontal axis is going to be cosine of x plus Delta x, and the projection over the vertical axis is going to be sine of x plus Delta x. Now let's focus on this orange triangle over here and let's actually blow it up a little bit. This triangle has one base equal to negative Delta cosine of x, and the other side is Delta sine of x. Now we can call the top angle Phi and the hypotenuse h. If you recall from trigonometry, there is a rule that states that given a right triangle, the cosine of one of the angles is equal to the adjacent side divided by the hypotenuse. The adjacent side here is Delta sine x, while the hypotenuse is h. Now given that Delta sine x equals h cos Phi. Similarly, there's a rule that states that the sine is equal to the opposite divided by the hypotenuse. In this triangle, the opposite side is negative Delta cos x, while the hypotenuse is again h. Therefore negative Delta cos h is h sine of Theta. Now what happens as delta gets smaller and smaller? Well, the hypotenuse of this triangle almost coincides with the arch of the circle and approaches Delta of x. The angle Phi approaches x and with this new values, we get that Delta sine x equals Delta x cos x, and negative cos x equals Delta x sine x. Also Delta sine x divided by Delta x goes to the derivative of sine x, and Delta cos x divided by Delta x is the derivative of cosine of x. This actually confirms the intuition you built before. Sine prime of x is cos x and cos prime of x is negative sine x.