Derivatives of Trig Functions

Derivatives of Trig Functions - Derivatives of Trig...

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Derivatives of Trig Functions With this section we’re going to start looking at the derivatives of functions other than polynomials or roots of polynomials. We’ll start this process off by taking a look at the derivatives of the six trig functions. Two of the derivatives will be derived. The remaining four are left to the reader and will follow similar proofs for the two given here. Before we actually get into the derivatives of the trig functions we need to give a couple of limits that will show up in the derivation of two of the derivatives. Fact See the Proof of Trig Limits section of the Extras chapter to see the proof of these two limits. Before we start differentiating trig functions let’s work a quick set of limit problems that this fact now allows us to do. Example 1 Evaluate each of the following limits. (a) [ Solution ] (b) [ Solution ] (c) [ Solution ] (d) [ Solution ] (e) [ Solution ]
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(f) [ Solution ] Solution (a) There really isn’t a whole lot to this limit. In fact, it’s only here to contrast with the next example so you can see the difference in how these work. In this case since there is only a 6 in the denominator we’ll just factor this out and then use the fact. [ Return to Problems ] (b) Now, in this case we can’t factor the 6 out of the sine so we’re stuck with it there and we’ll need to figure out a way to deal with it. To do this problem we need to notice that in the fact the argument of the sine is the same as the denominator ( i.e. both ’s). So we need to get both of the argument of the sine and the denominator to be the same. We can do this by multiplying the numerator and the denominator by 6 as follows. Note that we factored the 6 in the numerator out of the limit. At this point, while it may not look like it, we can use the fact above to finish the limit. To see that we can use the fact on this limit let’s do a change of variables . A change of variables is really just a renaming of portions of the problem to make something look more like something we know how to deal with. They can’t always be done, but sometimes, such as this case, they can simplify the problem. The change of variables here is to let and then notice that as we also have . When doing a change of variables in a limit we need to change all the x ’s into ’s and that includes the one in the limit. Doing the change of variables on this limit gives,
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And there we are. Note that we didn’t really need to do a change of variables here. All we really need to notice is that the argument of the sine is the same as the denominator and then we can use the fact. A change of variables, in this case, is really only needed to make it clear that the fact does work. [ Return to Problems
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This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.

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Derivatives of Trig Functions - Derivatives of Trig...

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