Trigonometric Derivatives - Part 2: Cosecant, Secant, Cotangent

Problem: We now have the derivative of sine, cosine, and tangent. Now we have the next set to find derivatives for.

If you haven't already read part 1, go ahead and read all about it here. If you have, go ahead and continue.

We now have the basic trigonometric derivatives.

Original -> Derivative

sin(x) -> cos(x)

cos(x) -> -sin(x)

tan(x) -> sec²(x)

Next, we need to find it for csc(x), sec(x), and cot(x). Luckily, this shouldn't take long, since they're all quotient rules, which was explained in the previous blog (the link is above). So we can easily apply them to these.

The next step will look weird, only because the value of 1 is in the numerator. Its derivative is 0, removing the first half of the derivative's numerator.

And finally, we can simplify.

Once again, that's a lot nicer than I thought. Next, we will find the derivative of sec(x), which should work out in a similar fashion.

This makes sense, considering the derivative of cos(x) makes us subtract a negative from 0, entering the positive realm. And because sin(x) was in the numerator and cos(x) in the denominator, it makes sense that tan(x) was the result instead of cot(x).

And finally, let's solve for the derivative of cot(x). 

I saw this, and for some reason thought it looked like it would be a pain. But really, we just end up with -1 in the numerator.

This is great! We have now found 6 of the trigonometric derivatives!

sin(x) -> cos(x)

cos(x) -> -sin(x)

tan(x) -> sec²(x)

csc(x) -> -csc(x)*cot(x)

sec(x) -> sec(x)*tan(x)

cot(x) -> -csc²(x)

In the next blog post, we will find the derivatives of arcsin(x), arccos(x), and arctan(x).