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### Course: AP®︎/College Calculus AB>Unit 2

Lesson 11: Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions

# Derivatives of sec(x) and csc(x)

Let's explore the derivatives of sec(x) and csc(x) by expressing them as 1/cos(x) and 1/sin(x), respectively, and applying the quotient rule. We discover that the derivative of sec(x) can be written as sin(x)/cos²(x) or tan(x)sec(x), and the derivative of csc(x) can be expressed as -cos(x)/sin²(x) or -cot(x)csc(x).

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• How can you prove the sex and cscx derivatives using the power rule? Because you can't take them as cos^-1x or sin^-1x, can you?
• You can prove the sec x and cosec x derivatives using a combination of the power rule and the chain rule (which you will learn later).
Essentially what the chain rule says is that
``d/dx(f(g(x)) = d/dg(x) (f(g(x)) * d/dx (g(x))``

When you have sec x = (cos x)^-1 or cosec x = (sin x)^-1,
you have it in the form f(g(x)) where f(x) = x^-1 and g(x) = cos x or sin x.
Below is the working for how to derive the derivatives of sec x using this:

``d/dx (sec x)= d/dx ((cosx)^-1)= -1 * (cos x)^-2 * d/dx (cos x)= -1 * (cos x)^2 * (-sin x)= sin x/(cosx)^2= sec x * tan x``

The same process can be repeated for cosec x and as can be clearly seen, the results are the same as what Sal gets.

Hope I helped!
• So, just by looking at the graphs, it is obvious that d/dx[sin(x)]=cos(x), & d/dx[cos(x)]=-sin(x). However, I'm just curious about the mathematical proof of it. Can somebody please redirect me to a video with the proof?
• For AP Calculus BC, is it expected that these 6 identities be memorized?
• In every calculus sequence I've seen, it was expected that you memorize the identities . Good luck!
• Use definition of derivative to show that
d÷dx(secx)=secx tanx
• 𝑓(𝑥) = sec 𝑥 = 1∕cos 𝑥 ⇒

⇒ 𝑓 '(𝑥) = lim(ℎ → 0) [(1∕cos(𝑥 + ℎ) − 1∕cos 𝑥)∕ℎ] =

= lim(ℎ → 0) [(cos 𝑥 − cos(𝑥 + ℎ))∕(ℎ cos 𝑥 cos(𝑥 + ℎ))] =

= lim(ℎ → 0) [(cos 𝑥 − cos 𝑥 cos ℎ + sin 𝑥 sin ℎ)∕(ℎ cos 𝑥 cos(𝑥 + ℎ))] =

= lim(ℎ → 0) [(1 − cos ℎ + tan 𝑥 sin ℎ)∕(ℎ cos(𝑥 + ℎ))] =

= lim(ℎ → 0) [(1 − cos ℎ)(1 + cos ℎ)∕(ℎ(1 + cos ℎ) cos(𝑥 + ℎ)) +
+ tan 𝑥 sin ℎ∕(ℎ cos(𝑥 + ℎ))] =

= lim(ℎ → 0) [sin²ℎ∕(ℎ(1 + cos ℎ) cos(𝑥 + ℎ)) + tan 𝑥 sin ℎ∕(ℎ cos(𝑥 + ℎ))] =

= lim(ℎ → 0) [(sin ℎ∕ℎ) ∙ (sin ℎ∕((1 + cos ℎ) cos(𝑥 + ℎ)) + tan 𝑥∕cos(𝑥 + ℎ)] =

= 1 ∙ (0∕(2 cos 𝑥) + tan 𝑥∕cos 𝑥) =

= tan 𝑥 sec 𝑥
• I always had a hard time trying to not mix up the concepts of the tangent function with concept of the tangent line. I have used a good deal of hours to try to understand the concepts and establish a relation between them and somehow it still isn't very easy for me to not mix it up sometimes.

Now I am presented with the function sec x, that I have missed from the geometry playlist, apparently. After getting this far in the calculus playlist I imediatly though of the sec x function having something to do with secant line but I fail to establish a clear relationship. I went to check the definition of sec x and I now know that it is the inverse of the cos x function but I can't help but try to make a relation with secant line without being able to in a satisfying manner. Any hint on how to think about this?
• It's unfortunate that "secant" and "tangent" serve double purposes in mathematics, but we're stuck with them now. This isn't really a calculus matter, it's trigonometry. There's a neat interactive diagram relating the six functions - sine, cosine, tangent, cotangent, secant, and cosecant - on this page:
In the "Appendix: All trig ratios in the unit circle" section. (About 2/5 down the page, just before the Q&A's.

You may be even able to see how tangent (the function) and a tangent line are related. I can see no obvious reason why the secant function is so named though.
• I'm a little confused overall because I thought the derivative of sec(u) was sec(u)tan(u)*du/dx. Is there a theorem or a law for the derivative of these trig functions? Thank you to whoever can answer!
• You didn't make any mistake. But I think the problem is that you have assumed u as a function of x. That's why by the chain rule you would get the du/dx part. In the video, Sal is only considering one variable x and has taken the derivative with respect to x. He has not taken a function of x. If he were to consider a function of x and then take the derivative of that function with respect to x then his answer would look similar to yours.
• Is it more efficient or beneficial to memorize the derivatives of sec(x), csc(x), tan(x) and cot(x) OR to spend extra time working through the proofs that Sal has just done for us here?
(1 vote)
• Personally, I'd advise you to practice problems with them. That way, you'll get used to using the derivatives and eventually, you'll remember them. Proofs are also good to know though, as they can help you derive the derivative incase you forget
• What means d/dx? That we need to find the derivative?
Is there a difference between dx/d and d/dx ?
Is dx/d suppose mean that we need to find integral?