Calculus! #35: Early Transcendentals 3.3

3.3: Derivatives of Trig Functions

The derivative of sinx is cosx. The derivative of cosx is -sinx. This leads you to a little 4 point circuit you can walk around whenever you take the derivative of one of these functions.

sinx

cosx

-sinx

-cosx

You should have this as working knowledge. It’ll make everything a lot easier. Once again, this is a section where you should work a lot of problems, just to make sure you have your head on straight about the above.

(Also, the precocious among you may notice a similarity between derivatives of sine and the function e^{ix}. Weird, right?)

Got it?

OKAY, so first the book goes into a common graphical demonstration of why this is intuitive. Remember, the derivative is graphically the slope of the tangent to a curve at a certain point. If you take a sine wave and figure the slope at a shitload of points, then plot those slope values on another graph, you’ll get cosine.

The book then gives a somewhat complicated geometric proof. Much like the radial acceleration proof we discussed in yesterday’s physics section, this proof is trickier than it needs to be, since there is a much prettier proof we’ll get to later in the book, using something called a Taylor Series. But, for your mathematical edification, it’s worth going over the proof in the book.

Once you accept that proof, the others follow fairly trivially.

The one thing I do want to go over, which is a bit of a reiteration of earlier sections about functions, is that all trig functions are just variations on sine. So, don’t be intimidated by the big list of derivatives. Each trig function can be deduced from sine, and thus their derivatives can be deduced using sine and some derivative rule (preferably the product rule or the chain rule, which you’ll get shortly).

So, for example, if you’re on a test and get asked the derivative of tanx, but can’t remember it offhand, just convert tanx to sinx/cosx and solve.

Next up: The Glorious Chain Rule.

 

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4 Responses to Calculus! #35: Early Transcendentals 3.3

  1. Jonathan says:

    It should be e^(ix), not e^(i) x

  2. David says:

    Related LaTeX tip: if you want a string of more than one character to appear in a superscript or subscript, you can enclose the thing in brackets.

    So to write the exponential of ix, you’d write e^{ix}. Let’s see if it renders down in the comments:

    $e^{ix}$

    If not, doesn’t really matter. But the way you have it, as Jonathan points out, looks like (e^i) * x.

    I enjoy that you call the Chain Rule “The Glorious Chain Rule.”

  3. I need the following math courses to get my degree:
    Calculus 1 or Applied Calculus 1 AND one of
    Calculus 2 or Applied Calculus 2
    But I don’t know the difference between the regular and applied calculus courses. Are they the same difficulty and do they have the same content?

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