we differentiated with respect to xxx and so when faced with xxx’s in the function we differentiated as normal and when faced with yyy’s we differentiated as normal except we then added a y′y′y' onto that term because we were really doing a chain rule.
The outside tells you what you need to differentiate in respect to.
we’ll leave the yyy’s written as yyy’s and in our head we’ll need to remember that they really are y(x)y(x)y\left( x \right) and that we’ll need to do the chain rule.
Make sure to remember which is which or else you might get lost within your solution and get it wrong.
e either won’t be able to solve for yyy
The main objective is to find the derived form of y, usually it will be complex.
we will often need to use the product or quotient rule for the higher order derivatives
As you move on to higher order derivatives it gets a little more complex, like you will have to use the product or quotient rules.
parenthesis in the exponent denotes differentiation while the absence of parenthesis denotes exponentiation.
You need to be careful about parenthesis, if you forget them you would be implying something different and your answer could be wrong.
we will get zero for all derivatives after this point.
Notice that after a certain point you will get to 0 and you will have to stop because there will be nothing to differentiate.
complicated examples
For complicated examples you can rewrite the problem to make it easier and then use the chain rule.
“inside function” and the “outside function”.
Have to make sure to identify both or else the derivative will be wrong.
simply won’t work to get the derivative here.
Once there is a function within a function you usually need more than one thing to find the derivative.
recalling the definition of the inverse sine function.
You need to manipulate the equations into something that will be more understandable.
there are in fact an infinite number of angles that will work
You need to follow the restriction or else there will be too many answers to choose from.
no restrictions on xxx because tangent can take on all possible values.
Unlike sine tangent doesn't have any restriction because it can be any number. Therefore we don't have to worry about it falling within a certain quadrant.
Remember that in these notes we tend to take positive angles
Usually, working with positive angles is better and easier, so converting them to positive angles is the best thing to do.
Note that if we’d canceled the xxx we would have missed the first solution.
Unlike many things that were taught to us before, here the possibility of a different answer for x shouldn't be ignored, not even when you factor.
in this case the “-” are coming about when we solved for xxx after computing the inverse cosine in our calculator.
Sometimes the angle might be negative but that just means that it's moving clockwise instead of counterclockwise. The negative and positive give us direction in where the angle is at.
our calculator only gave us a single answer.
The reason why this happens is because the calculator has certain restrictions, which is why we're doing all the other work in the firs place.
two different notations that are commonly used
The inverse of cosine, sine and tangent is one of the methods, which seems like the simplest way to do it. All that is needed is to integrate that into the steps we learned before.
many calculators can’t handle inverse secant so we’re going to need a different solution method for this one
There needs to be different changes in order to solve for t. You need to know different things, like that secant is the opposite of cosine, so you can plug it in the calculator.
since cosine will never be greater that 1 it definitely can’t be 2.
Probably because the hypotenuse is equal to 1.
Unlike the previous example only one of these will be in the interval.
The only intervals that count are the ones that fall in the domain. If they are greater than or smaller then they can't be part of the solution.
We are looking for values of xxx so divide everything by 5 to get.
What you do to one side, you have to do to the rest to keep the same values.
emember that we’re typically looking for positive angles between 0 and 2π2π2\pi so we’ll use the positive angle.
When subtracting then 2π has to be in the front so the angles can be positive. Ex: 2π minus π/3