So far, we've learned our derivatives for our sine and cosine functions. But we know that there are way more trig functions than just sine and cosine, and we'll need to know the derivatives of those trig functions as well. And that's exactly what we're going to look at here. Here, I'm going to show you all of the derivatives for our remaining trig functions as well as where they come from. And we'll also work through some examples putting all of the rules that we've learned so far for taking derivatives together.

So let's go ahead and jump in here. Now we saw that when taking the derivatives of our sine and cosine functions their derivatives were just other trig functions, and this is going to be true for all of our trig functions. All of their derivatives are just some variation or combination of other trig functions like here with the tangent of x. Now, I'm just going to go ahead and give this derivative to you. The derivative of the tangent of x is equal to the secant squared of x.

That's just another trig function. Now, it's going to be really useful for you to just go ahead and memorize all of your derivatives for your trig functions, but it may also be helpful to know where these come from. How do we know that the derivative of the tangent of x is the secant squared of x? Well, when working with any of these other trig functions, that's just trig functions that aren't sine or cosine, we can actually rewrite all of them in terms of sine and cosine and find their derivatives using the quotient rule. So let's backtrack a little bit and go ahead and derive this derivative for the tangent of x.

Now we know that we can rewrite the tangent of x as the sine of x over the cosine of x. So since I have one function being divided by another, I can find my derivative using the quotient rule. Now we know that the quotient rule tells us low d high minus high d low. That's our numerator here. So I have my bottom function, that low function, cosine, multiplied by the derivative of that high function.

Now the derivative of the sine of x is the cosine of x. And then I'm subtracting here high d low, so my high function at sine times the derivative of this bottom function cosine. The derivative of cosine x is negative sine of x. And then all of this is divided by the square of what's below. So we took that bottom function cosine and squared it.

Now from here if I multiply out this numerator fully I end up getting the cosine squared of x plus the sine squared of x and keeping that denominator the same the cosine squared of x. Now I notice here that I have a trig identity because the cosine squared of x plus the sine squared of x is just equal to 1. So that leaves me with 1 over the cosine squared of x, which I know can simplify to the secant squared of x. So that's exactly how we get our derivative of the tangent of x. Now we could follow this same exact process for all of our remaining trig functions.

That's the cotangent, the secant, and the cosecant. And you can feel free to go through that process on your own. But here we're going to work through all of these remaining derivatives in practice and take a look at some examples combining all of the rules that we've learned so far for taking derivatives with our derivatives of our trig functions. So let's dive into this first example here. Here, we're asked to find the derivative of f of x is equal to 3x squared plus the cotangent of x.

Now here I want to use the sum rule because I have 2 functions being added together. So I know that in order to find my derivative here f prime of x, I just need to take the derivative of each of these individual functions and add them together. Now for that first term 3x squared, I can easily find that derivative using the power rule. I know that my derivative here is just going to be a 6x. Then I need to add this together with the derivative of the cotangent of x.

So let's come over to our table here and take a look at our derivative of cotangent. The derivative of cotangent is going to be the negative cosecant squared of x. And looking at all of these cofunctions, it may be helpful in memorizing these to know that all of the derivatives of our cofunctions will always be negative. The derivative of cosine is negative sine. The derivative of cotangent is negative cosecant.

And the derivative of cosecant is negative cotangent of x times the cosecant of x. So let's go ahead and apply this derivative of cotangent in our problem over here. So we're going to add this together with the negative cosecant squared of x. That's my derivative of cotangent. And I can, of course, rewrite this a little bit as 6x minus the cosecant squared of x for this final derivative.

And this is my answer here. So let's move on to our next example. Here, we want to find the derivative of f of x is equal to 4x times the secant of x. Now here, I have 2 functions being multiplied. I have 4x, and I have the secant of x.

So that means I need to use the product rule in order to find my derivative here, f prime of x. Now remember that our product rule tells us left d right plus a right d left. So I'm gonna start by taking that left-hand function for x and multiplying it by the derivative of that right-hand function. So I want to take the derivative here of the secant of x. Now coming back up to our table here, I can see that the derivative of the secant of x is equal to the tangent of x times the secant of x.

So that's exactly what I'm going to write here. Tangent x times secant x. That's my derivative of this right-hand function secant x. Then continuing on with my product rule here, I am adding this together with right d left So that right-hand function as is the secant of x multiplied by the derivative of that left-hand function. So here the derivative of 4x, we know that that is just equal to 4, and we've successfully applied our product rule here.

Now we can do a little bit of simplifying because there are some common terms here. I have 4 and I have secant x in both of these terms. So I can go ahead and factor that out, giving me my derivative here f prime of x is equal to 4 secant x, having factored that out here multiplied by x times the tangent of x plus 1. And this is my final answer here for this derivative. Now like I mentioned earlier it's going to be most useful to go ahead and memorize all of your derivatives for trig functions.

But if you ever forget any of them, remember that you can follow the process that we did above, rewriting in terms of sine and cosine and using the quotient rule to find that derivative. But, again, the most useful thing here is to memorize them because you're gonna be working with them a ton. Now let's get some practice with these trig functions coming up next. I'll see you there.