Derivatives of sin(x) and cos(x) | Derivative rules | AP Calculus AB | Khan Academy

What I'd like to do in this video is get an intuitive sense for what the derivative with respect to x of sine of x is and what the derivative with respect to x of cosine of x is. I've graphed y is equal to cosine of x in blue and y is equal to sine of x in red. We're not going to prove what the derivatives are, but we're going to know what they are and get an intuitive sense. In future videos, we'll actually do a proof.

So let's start with sine of x. The derivative can be viewed as the slope of the tangent line. So for example, at this point right over here, it looks like the slope of our tangent line should be zero. So our derivative function should be zero at that x value. Similarly, over here, it looks like the derivative is zero; the slope of the tangent line would be zero. So whatever our derivative function is at that x value, it should be equal to zero.

If we look right over here on sine of x, it looks like the slope of the tangent line would be pretty close to 1. If that is the case, then in our derivative function, when x is equal to 0, that derivative function should be equal to one. Similarly, over here, it looks like the slope of the tangent line is negative one, which tells us that the derivative function should be hitting the value of negative one at that x value.

So you're probably seeing something interesting emerge everywhere. While we’re trying to plot the slope of the tangent line, it seems to coincide with y is equal to cosine of x. And it is indeed the case that the derivative of sine of x is equal to cosine of x. You can see that it makes sense, not just at the points we tried, but even in the trends. If you look at sine of x here, the slope is one, but then it becomes less and less positive all the way until it becomes zero.

Cosine of x, the value of the function is one, and it becomes less and less positive all the way until it equals zero. You could keep doing that type of analysis to feel good about it. In another video, we're going to prove this more rigorously.

So now let's think about cosine of x. Cosine of x right over here, the slope of the tangent line looks like it is zero, and so its derivative function needs to be zero at that point. So hey, maybe it's sine of x. Let's keep trying this.

So over here, cosine of x looks like the slope of the tangent line is negative one, and so we would want the derivative to go through that point right over there. All right, this is starting to seem; it doesn't seem like the derivative of cosine of x could be sine of x. In fact, this is the opposite of what sine of x is doing. Sine of x is at one, not negative one at that point. But that's an interesting theory: maybe the derivative of cosine of x is negative sine of x.

So let's plot that. So this does seem to coincide. The derivative of cosine of x here looks like negative one, the slope of the tangent line, and negative sine of this x value is negative one. Over here, the derivative of cosine of x looks like it is zero, and negative sine of x is indeed zero.

So it actually turns out that it is the case that the derivative of cosine of x is negative sine of x. So these are really good to know. These are kind of fundamental trigonometric derivatives to know. We'll be able to derive other things for them, and hopefully, this video gives you a good intuitive sense of why this is true. In future videos, we will prove it rigorously.