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• - [Voiceover] We already know the derivatives

• of sine and cosine.

• We know that the derivative with respect to x

• of sine of x is equal to cosine of x.

• We know that the derivative with respect to x

• of cosine of x is equal to negative sine of x.

• And so what we want to do in this video

• is find the derivatives of the other basic trig functions.

• So, in particular, we know, let's figure out

• what the derivative with respect to x,

• let's first do tangent of x.

• Tangent of x, well this is the same thing

• as trying to find the derivative with respect to x of,

• well, tangent of x is just sine of x,

• sine of x over cosine of x.

• And since it can be expressed

• as the quotient of two functions,

• we can apply the quotient rule here to evaluate this,

• or to figure out what this is going to be.

• The quotient rule tells us that this is going to be

• the derivative of the top function,

• which we know is cosine of x times the bottom function

• which is cosine of x, so times cosine of x

• minus, minus the top function, which is sine of x,

• sine of x, times the derivative of the bottom function.

• So the derivative of cosine of x is negative sine of x,

• so I can put the sine of x there,

• but where the negative can just cancel that out.

• And it's going to be over, over

• the bottom function squared.

• So cosine squared of x.

• Now, what is this?

• Well, what we have here, this is just a cosine squared of x,

• this is just sine squared of x.

• And we know from the Pythagorean identity,

• and this is really just out of,

• comes out of the unit circle definition,

• the cosine squared of x plus sine squared of x,

• well that's gonna be equal to one for any x.

• So all of this is equal to one.

• And so we end up with one over cosine squared x,

• which is the same thing as, which is the same thing as,

• the secant of x squared.

• One over cosine of x is secant,

• so this is just secant of x squared.

• So that was pretty straightforward.

• Now, let's just do the inverse of the,

• or you could say the reciprocal, I should say,

• of the tangent function, which is the cotangent.

• Oh, that was fun, so let's do that,

• d dx of cotangent,

• not cosine, of cotangent of x.

• Well, same idea, that's the derivative with respect to x,

• and this time, let me make some sufficiently large brackets.

• So now this is cosine of x over sine of x,

• over sine of x.

• But once again, we can use the quotient rule here,

• so this is going to be the derivative of the top function

• which is negative, use that magenta color.

• That is negative sine of x

• times the bottom function,

• so times sine of x, sine of x,

• minus, minus

• the top function, cosine of x,

• cosine of x, times the derivative of the bottom function

• which is just going to be another cosine of x,

• and then all of that over the bottom function squared.

• So sine of x squared.

• Now what does this simplify to?

• Up here, let's see, this is sine squared of x,

• we have a negative there,

• minus cosine squared of x.

• But we could factor out the negative

• and this would be negative sine squared of x

• plus cosine squared of x.

• Well, this is just one by the Pythagorean identity,

• and so this is negative one over sine squared x,

• negative one over sine squared x.

• And that is the same thing as

• negative cosecant squared,

• I'm running out of space, of x.

• There you go.

- [Voiceover] We already know the derivatives

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# Derivatives of tan(x) and cot(x) | Derivative rules | AP Calculus AB | Khan Academy

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yukang920108 posted on 2022/07/12
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