If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Proof of the derivative of cos(x)

## Video transcript

what I wanted 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 negative sine of X and we're going to base this argument based on a previous proof we made that the derivative with respect to X of sine 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 sine of X so right over here we see sine of X in red and we see cosine of X in blue and 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 and 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 2 shift it to the left by PI over 2 and I'm also going to shift the blue graph to the left by PI over 2 and so what am I going to get well the blue graph is going to look like this one right over here and if it was cosine of X up here we can now say that this is equal to Y is equal to cosine of X plus PI over 2 this is the blue graph cosine of X shifted to the left by PI over 2 and this is y is equal to sine of X plus PI over 2 now the visual argument is all I did is I shifted both of these graphs to the left by PI over 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 sine of X plus PI over 2 that that is equal to the blue graph that that is equal to cosine of X plus I over to now what is sine of X plus PI over 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 and you can also see it intuitively or graphically just by looking at these graphs and what is cosine of X plus PI over 2 well once again from our trig identities we know that that is the exact same thing as negative sine of X so there you have it the visual argument just start with this knowledge shift both of these graphs to the left by PI over 2 it should still be true that the derivative with respect to X of sine of X plus PI over 2 is equal to cosine of X plus PI over 2 and 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 the cosine of X in this video
AP® is a registered trademark of the College Board, which has not reviewed this resource.