Question

Calculate the derivatives of the following real functions by using the chain rule; maybe several times. In each case, determine the inner and outer function to be used, together with their respective areas of definition.
b) \(h(x) = exp(\frac{1}{sin x})\)

Answer 1

First, I would write it as \[\large h(x)=e^{\frac{1}{\sin(x)}}\]Now, can you tell me what the outer function is?

Answer 2

i have no idea to be honest, its also important for mx exam, which part of math should i learn, can you send me maybe a link George?

Answer 3

i know i must memorize some stuff for derivatives..

Answer 4

Looking at this, you can tell that the outer function is \(e^u\) if you let \(u=\frac{1}{\sin(x)}\). Recall that the derivative of \(e^u\) is \(e^u\) itself. That means you're looking for \[\large \frac{d}{dx}\;e^{\frac{1}{\sin(x)}}=e^{\frac{1}{\sin(x)}}\cdot \frac{d}{dx}\;\frac{1}{\sin(x)}\]

Answer 5

ok..

Answer 6

Now rewrite \[\frac{1}{\sin(x)}=\sin^{-1}(x).\]We now apply the chain rule again, this time using the power rule to differentiate. We get \[\large e^{\frac{1}{\sin(x)}}\cdot \frac{d}{dx}\;\sin^{-1}(x)=e^{\frac{1}{\sin(x)}}\cdot -\sin^{-2}(x)\cdot \frac{d}{dx}\;\sin(x)\]

Answer 7

We have one more step to go. We just need to take the derivative of \(\sin(x)\). That means the final derivative should be \[\large e^{\sin^{-1}(x)}\cdot -\sin^{-2}(x)\cdot \frac{d}{dx}\;\sin(x)= e^{\sin^{-1}(x)}\cdot -\sin^{-2}(x)\cdot\cos(x)\]We can rewrite this in fraction form to be \[\large e^{\frac{1}{\sin(x)}}\cdot -\frac{\cos(x)}{\sin^{2}(x)}\]

Answer 8

ok there is a lot ofs thing to memorize learn for me i think, how can i search for this thema what is the name of this topic ?

Answer 9

In general, the best way to memorize these is to do them. However, there is some useful information here http://en.wikipedia.org/wiki/Table_of_derivatives and here http://en.wikipedia.org/wiki/Derivative#Computing_the_derivative

Answer 10

ok this is exact what i want, i need to make me warm slowly for exam..

Answer 11

We're not done with the problem yet however. We still need to list the functions we looked at, and they're respective domains.

The first function (the outer function) is just \(e^x\). This has a domain of all reals. The next is the function \(\frac{1}{x}\). This has a domain of all reals except 0. The final function was \(\sin(x)\) which has a domain of all reals. Only the first function is an outer function. The rest are inner functions.

Answer 12

ok..

Answer 13

pls feel free to go on

Answer 14

That was the rest of it right there.

Answer 15

aah ok thank you George :) sorry

Answer 16

You're welcome.