Proof of the derivative of cos(x) | Derivative rules | AP Calculus AB | Khan Academy

What I'm going to do in this video is make a visual argument as to why the derivative with respect to X of cosine of x is equal to sin of X. We're going to base this argument on a previous proof we made that the derivative with respect to X of sin of X is equal to cosine of x. So, we're going to assume this over here. I encourage you to watch that video; that's actually a fairly involved proof that proves this. But if we assume this, I'm going to make a visual argument that this right over here is true: that the derivative with respect to X of cosine of x is negative sin of X.

So, right over here, we see s of X in red, and we see cosine of x in blue. We're assuming that this blue graph is showing the derivative, the slope of the tangent line for any x value of the red graph. We've gotten an intuition for that in previous videos. Now, what I'm going to do next is I'm going to shift both of these graphs to the left by pi over two. Shift it to the left by pi over two, and I'm also going to shift the blue graph to the left by Pi / 2.

And so, what am I going to get? Well, the blue graph is going to look like this one right over here. If it was cosine of x up here, we can now say that this is equal to Y is equal to sine of x + pi / 2. This is the blue graph cosine of x shifted to the left by Pi / 2, and this is y is equal to S of x + pi / 2.

Now, the visual argument is all I did is I shifted both of these graphs to the left by Pi / 2, so it should still be the case that the derivative of the red graph is the blue graph. So, we should still be able to say that the derivative with respect to X of the red graph s of x + pi / 2, that that is equal to the blue graph, that that is equal to cosine of x + Pi / 2.

Now, what is sin of x + Pi / 2? Well, that's the same thing as cosine of x. You can see this red graph is the same thing as cosine of x; we know that from our trig identities. You could also see it intuitively or graphically just by looking at these graphs. And what is sine of x + pi / 2? Well, once again, from our trig identities, we know that that is the exact same thing as negative sin of X.

So there you have it: the visual argument just starts with this knowledge, shifts both of these graphs to the left by Pi / 2, and it should still be true that the derivative with respect to X of sine of x + Pi / 2 is equal to cosine of x + Pi / 2. This is the same thing as saying what we have right over here. So now we should feel pretty good; we proved this in a previous video, and we have a very strong visual argument for this in this for cosine of x in this video.