Derivatives of sec(x) and csc(x) | Derivative rules | AP Calculus AB | Khan Academy

In a previous video, we used the quotient rule in order to find the derivatives of tangent of X and cotangent of X. What I want to do in this video is to keep going and find the derivatives of secant of X and cosecant of X. So, let's start with secant of X.

The derivative with respect to X of secant of X, well, secant of X is the same thing as 1 over cosine of X. That's just the definition of secant. You could, there are multiple ways you could do this. When you learn the chain rule, that actually might be a more natural thing to use to evaluate the derivative here. But we know the quotient rule, so we will apply the quotient rule here. It's no coincidence that you get to the same answer; the quotient rule actually can be derived based on the chain rule and the product rule. But I won't keep going into that. Let's just apply the quotient rule right over here.

So this derivative is going to be equal to the derivative of the top. Well, what's the derivative of one with respect to X? Well, that's just zero times the function on the bottom, so times cosine of X. Cosine of X minus the function on the top, well, that's just one times the derivative on the bottom. Well, the derivative of the bottom, the derivative of cosine of X, is negative sine of X. So, we could put the sine of X there, but it's negative sine of X. So you have a minus, and there would be a negative, so we could just make that a positive.

Then all of that over the function on the bottom squared, so cosine of X squared. And so, zero times cosine of X, that is just zero. All we are left with is sine of X over cosine squared of X.

There's multiple ways you could rewrite this if you like. You could say that this is the same thing as sine of X over cosine of X times 1 over cosine of X. Of course, this is tangent of X times secant of X. So, you could say the derivative of secant of X is sine of X over cosine squared of X, or it is tangent of X times secant of X.

So now let's do cosecant. The derivative with respect to X of cosecant of X, well, that's the same thing as the derivative with respect to X of 1 over sine of X. Cosecant is 1 over sine of X. I remember that because you think it's cosecant; maybe it's the reciprocal of cosine, but it's not. It's the opposite of what you would expect. The cosine reciprocal isn't cosecant; it is secant.

Once again, it’s just the way it happened to be defined. Anyway, let's just evaluate this. Once again, we'll do the quotient rule, but you could also do this using the chain rule.

So, it's going to be the derivative of the expression on top, which is zero, times the expression on the bottom, which is sine of X. Sine of X minus the expression on top, which is just one, times the derivative of the expression on the bottom, which is cosine of X.

All of that over the expression on the bottom squared, sine squared of X. That's zero, so we get negative cosine of X over sine squared of X. So, that's one way to think about it. Or if you like, you could view this the same way we did over here: this is the same thing as negative cosine of X over sine of X times 1 over sine of X, and this is negative cotangent of X.

Negative cotangent of X times, let me write it this way: times 1 over sine of X is cosecant of X. Cosecant of X. So, whichever one you find more useful.