Question

I'm looking for the first derivative of

f(x)=csc^2(sqrtx)

basically how do i get rid of the csc^2

Answer 1

can i do csc x csc(sqrtx) ?

Answer 2

let put that better csc(csc(sqrt(x)))

Answer 3

You have to use the chain rule. The derivative of csc(x)=-csc(x) cot(x)
\[\csc(x^{\frac{1}{2}})(-\csc(x^{\frac{1}{2}}) \cot(x^{\frac{1}{2}}))(\frac{1}{2}x^{-\frac{1}{2}})\]

Answer 4

what is that though the multiplication rule?!?!

Answer 5

Through the chain rule. First you deal with f(____)=____^2

Answer 6

csc^2

Answer 7

Oh, shoot, I forgot the 2.

Answer 8

It should be 2csc___

Answer 9

ok i see thats just the derivative of csc^2 right?

Answer 10

Right. And you leave the inside the same. Then you take the derivative of csc next.

Answer 11

but why do you take the derivative twice there? the chain rule kind of confuses me...

Answer 12

Think of it this way. You have a composition of functions:
\[f(x)=x^2\]\[g(x)=\csc(x)\]\[h(x)=x^{\frac{1}{2}}\]

When you combine them, you get
\[f(g(h(x)))=(\csc(x^{\frac{1}{2}}))^2\]

Answer 13

That's what your original function was. All you have to do is take the derivative of each function individually.
The first thing we did was take care of f(x). We have to take the derivative of csc next because that's g(x)

Answer 14

so,
f(s)=2x
g(x)=-csc(x)cot(x)
h(x)=1/2x^-1/2

Answer 15

so -2xcsc(1/2x^-1/2)cot(1/2x^-1/2)

Answer 16

and thats my final answer?!?!

Answer 17

Close. The only catch is that you're not taking f(x). You're taking\[f(\csc(x^{\frac{1}{2}})\]
That means that you don't have 2(x). You have \[2(\csc(x^{\frac{1}{2}}))\]

Later on, you're not taking g(x). You're taking \[g(x^{\frac{1}{2}})\]
That means you don't have csc(x)cot(x). You have
\[\csc(x^{\frac{1}{2}})\cot(x^{\frac{1}{2}})\]

Answer 18

Forgot a - in the csc part

Answer 19

ok i can work with that

Answer 20

so just add negative sign up there?

Answer 21

what what about the 1/2x^-1/2?

Answer 22

Final answer should look like
\[2\csc(x^{\frac{1}{2}})(-\csc(x^{\frac{1}{2}})\cot(x^{\frac{1}{2}}))(\frac{1}{2}x^{-\frac{1}{2}})\]
That simplifies to
\[-\frac{1}{x^2} \csc^2 (x^{\frac{1}{2}})\cot(x^{\frac{1}{2}})\]

Answer 23

The part with the 1/2x^(-1/2) was good because you were actually taking h(x).

Answer 24

but why isn't that in our final answer?

Answer 25

Which one? The one I wrote up above was an unsimplified version.

Answer 26

the 1/2x^-1/2, and my online homework said that the simplified answer up there was incorrect

Answer 27

I'm pretty sure it's correct. Maybe it wants the less simplified version? What did you enter?

Answer 28

the simplified let try the less simplified one

Answer 29

apparently that was it man that did it.

Answer 30

Oh good. It's unfortunate that it wouldn't accept a more simplified version.