[music] We've seen how to differentiate sine, and cosine and tangent, but how do I
differentiate secant? So let's differentiate secant.
And one way to get a handle on secant is to write secant as a composition of two
different functions. If I set f of x equals 1 over x, and g of
x equals cosine x. Then f of g of x is secant x.
So if I want to differentiate secant, it'd be good enough to differentiate f of g of
x. And I can do that differentiation problem
by using the chain rule. Now I can separately calculate the
derivative of 1 over, it's negative 1 over x squared.
And the derivative of cosine is negative sine.
So putting this together, the derivative of secant must be the derivative of f,
which is minus 1 over its input squared at g of x and g is cosine.
So, with negative 1 over cosine square that's f prime of g of x times the
derivative of g which is minus sine. Now I can put this together, the minus
signs cancel and I'm left with sine x over cosine squared x.
But people don't usually write it this way I could instead write this, as sine x over
cosine x times 1 over cosine x. In other words, I could write this as,
what's this term, this is tangent and this is secant so I can write this as tangent x
secant x. So the derivative of secant x is tangent x
times secant x. We can play the same kind of game to
differentiate cosecant. So what about the derivative of cosecant
x? Well think about how I can rewrite
cosecant, alright? Cosecant is the composition of the one
over function and sine, right? Cosecant is one over sine.
So this is the derivative of f of g of x, where f is 1 over and g is sine, and by
the chain rule, that's the derivative of f at g times the derivative of g.
And now I know what the derivatives of these functions are.
The derivative of 1 over is minus 1 over x squared, and the derivative of g is cosine
x. And consequently, the derivative of
cosecant x must be minus 1 over, that's the derivative of f at g of x which is
sine, so it's sine squared times the derivative of g, which is cosine x.
And I can combine these to get minus cosine x over sine x times 1 over sine x.
In other words, minus cosine over sine, that's minus cotangent x, and minus 1 over
sine, well that's cosecant again. So the derivative of cosecant x is minus
cotangent x times cosecant x. And we can complete the story by
differentiating cotangent. Okay, so how do I differentiate cotangent
x ? Well we actually have some choices.
One way would be to write this as the derivative of cosine x over sine x, since
cotangent is cosine over sine. And then about the quotient rule.
Or, I could use the fact that cotangent is one over tangent, and then differentiate
this using the chain rule. To differentiate 1 over something, that's
negative 1 over the thing squared times the derivative of the inside function
which is tangent x. And I know the derivative of tangent x
that's secant squared, so this is negative 1 over tangent squared x times secant
squared x and that means the derivative of cotangent is negative secant squared x
over. Tangent squared x.
But people don't usually write it like this.
Instead, I could simplify this. Having a secant squared in the numerator
is as good as having a cosine squared in the denominator, and having a tangent in
the denominator, Is as good as having cosines, in the numerator, and sines in
the denominator. So, a negative secant squared over tangent
squared is the same as negative cosine squared over cosine squared times sine
squared. Now, these cosines cancel And what I'm
left with is negative 1 over sine squared x.
But people don't even write it this way, right?
1 over sine, well that's cosecant so this is actually negative cosecant squared x.
Now that we've seen the derivatives of all six of our trig functions, how are we
suppose to remember these derivatives? Well here's a table.
Showing all the derivatives of these 6 trig functions.
And you can see there's some real pattern to this table.
The derivative of sine is cosine. The derivative of cosine has a negative
sine. The derivative of tangent we saw is secant
squared. And the derivative of cotangent is
cosecant but with a negative sine. And the derivative of secant we just saw
is secant tangent. And the derivative of cosecant is negative
cosecant cotangent. So there's definitely some symmetry here,
and you can exploit that symmetry to help you to remember these derivatives.