# Derivatives of Inverse Hyperbolic Composite Trig Functions

## Метаданные

- **Канал:** The Organic Chemistry Tutor
- **YouTube:** https://www.youtube.com/watch?v=Xrlb3HlxX1A
- **Источник:** https://ekstraktznaniy.ru/video/41227

## Транскрипт

### Segment 1 (00:00 - 03:00) []

Number 92. Find dy over dx. So we want to find the derivative of a composite inverse hyperbolic trig function. To do that we need to know the formula. And here it is. The derivative of the inverse of the hyperbolic sine function of u, it's going to be u prime over the square<unk> of u ^2 + 1. For inverse hyperbolic cosine, it's u prime over the square<unk> of u ^2us 1. So in this problem, u is the inner function tangent x u prime the derivative of tangent will be sec^ 2 x. So now all we need to do is plug it into this formula. dy over dx is going to be u prime which is secant^ squ and that's going to be over the square t of u ^2 u is tangent so u ^2 will be tangent squ and then + 1. Now that is the answer but we could simplify it. Now you need to be familiar with certain trigonometric identities. Sin^ 2 + cosine^ 2 is equal to 1. 1 + tangent squ is equal to secant^ squ and 1 + 2 is equal to coseant squ. So for these kinds of problems you just need to know these trig identities. when you took trig. You can always just print the formula sheet for that. if you go to YouTube and you type in trigonometry for beginners organic chemistry tutor, you'll see a trick video come up. And if you download one of the free formula sheets that I have, you'll find these formulas there. So what we can do in this problem is we can replace 1 + tangent^ 2 or tan^ 2 + 1 with secant^ squ. The square root of squ is secant and secant^ 2 / secant. If you subtract the exponents 2 minus one, you're going to be left with seeant. And so this is going to be the final answer. The derivative of the inverse, the hyperbolic sine inverse of tangent x, that's going to equal secant x.