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# Derivatives of sin(x) and cos(x)

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.4 (EK)

## Video transcript

what I'd like to do in this video is going to 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 and 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 getting into a descent and then future videos will actually do a proof so let's start with sine of X so 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 one if that is the case then in our derivative function when X is equal to zero 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 and 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 1 but then it becomes less and less and less positive all the way until it becomes 0 cosine of X the value of the function is 1 and it becomes less and less positive all the way until it equals 0 and 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 so cosine of X right over here the slope of tangent line looks like it is zero and so it's 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 it 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 alright 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 1 not negative 1 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 1 the slope of the tangent line and negative sine of this x value is negative 1 over here the derivative of cosine of X looks like it is 0 and negative sine of X is indeed 0 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 and in future videos we will prove it rigorously
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