Derivative of cos(x)

by Batool Akmal

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    00:01 So if you enjoyed the proof for sin x, now is your chance to take the lead on the proof for cos x.

    00:07 It?s very similar to what we?ve just done with the same methods.

    00:10 It?s incredible that we?ve learnt these derivatives all through school but we?ve never really seen where they come from and now is your chance to actually prove it to yourselves.

    00:19 So, using exactly the same method, let?s now find the derivative for cos x.

    00:25 Remember, we kind of, not kind of we do know the answer to this already but let?s see if we can actually derive it algebraically.

    00:32 So, once again using differentiation from first principles, we are saying that dy/dx as the limit of delta x tends to zero is f of x plus delta x minus f of x divided by delta x. This time, we are differentiating cos x or we?re finding what the derivative of cos x is.

    00:54 So, that?s what we?re trying to do as the limit of delta x tends to zero.

    00:59 So, we can rewrite this as cos x plus delta x minus the original cos x all over delta x.

    01:09 This functional now, we will need to expand using our double angle or addition law formulas that we gave right at the start.

    01:19 So, remember that you can compare this with cos of a plus b.

    01:24 So this is the same as cos of a plus b and you can look that rule up and actually use that to expand cos x plus delta x.

    01:33 So, if we expand this using our rule, we can say cos x cos delta x.

    01:39 This time we have a minus sin x sin delta x minus cos x over delta x is our differentiation from first principles. Again, we can now combine our cos x terms.

    01:56 So you have a cos x value here and a cos x value here.

    01:59 So we can just rewrite it in order to write them next to each other so it doesn?t look too confusing.

    02:04 So we have cos x delta x minus cos x and then you also have minus sin x sin delta x.

    02:13 And don?t forget that this is all over delta x.

    02:19 The reason we?ve done that is so we can rewrite it as a common factor.

    02:24 So you can take cos x out of the first 2 terms. So, those 2 terms there, leaving you with cos of delta x minus 1 over delta x, and then you can take sin x out, leaving you with sin delta x over delta x. And again, don?t forget, right at the start we said that the limit of delta x should tend to zero.

    02:53 So for each part we can now apply it to each individual, let?s just put delta x here so we can follow what we?re saying.

    03:04 So for each individual section of this equation, we now apply delta x equals to zero.

    03:09 Let?s do that. Cos x isn?t really affected because there?s no delta x there. So, we have cos x.

    03:14 This if you remember from the previous proof that cos of delta x as delta x tends to zero, this goes to 1. Okay? Because cos of, as you get closer and closer to zero is 1.

    03:27 So, 1 minus 1 gives you zero. So this entire term becomes zero.

    03:31 Here, we have minus sin x which isn?t affected by delta x and remember again, from our very first statement on one of the properties that we mentioned earlier, that sin x over x as the limit of x tends to zero is 1.

    03:46 Our x in this case is delta x, so it?s the same thing, it?s still 1.

    03:50 So that multiples with 1 and eventually, we can now say that dy/dx of this function is minus sin x and here we have derived the differential or the standard differential answer to cos of x.

    04:06 So we said at the beginning that cos of x equals to minus sin x and we?ve proven now.

    About the Lecture

    The lecture Derivative of cos(x) by Batool Akmal is from the course Differentiation of Trigonometric Functions.

    Included Quiz Questions

    1. -sin(x)
    2. sin(x)
    3. -cos(x)
    4. tan(x)

    Author of lecture Derivative of cos(x)

     Batool Akmal

    Batool Akmal

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