Question

How do I find the derivative of y with respect to the appropriate value:
y=cos−1(x^2)

Answer 1

the derivative of y with respect to x?

Answer 2

yes I believe so

Answer 3

is the original equation
y=cos(-1x^2)

Answer 4

inverse of cosine
so y=cos^1(x^2)

Answer 5

\[\cos ^{-1}(x ^{2})\]
that makes sense now
do you know the chain rule?

Answer 6

not much. Thats why I really need help with this

Answer 7

the chain rule is a way to find derivatives of composite functions
example
you could find the derivative of (2x+3)^2 by multiplying out the entire thing and using the power rule
or you could use the chain rule
the chain rule says that you can substitute in "u" for part of an expresion, take the derivative of the new expression, then multiply it by the derivative of "u"
in our example, u=(2x+3)
the new equation is u^2
derivative is 2u
derivative of u is 2
multiply the derivative of the new equation by the derivative of u and substituting gives you 2(2x+3)(2)=8x+12
if you multiply out the polynomial and use the power rule, you get the same thing
try it and see

Answer 8

so it would be cos^1(2x)?

Answer 9

you can substitute u=x^2
derivative of u = 2x
derivative of cos^-1(u) is -1/(sqrt(1-u^2)
if you multiply that by the derivative of u and then re-substitute, you get 2x/(1-sqrt(x^4))
try it and see what you get

Answer 10

oh okay thank you!

Answer 11

actually, let me draw out what it should be because my answer actually had another mistake in it

Answer 12

thank you so much!

Answer 13

do you need more examples?

Answer 14

well what about for, y=sin^-191-t) ? this one is more complicated.

Answer 15

sin^-1(191-t)

Answer 16

i mean y=sin^-1 (1-t) sorry

Answer 17

what can you set u equal to to make this easier (hint: usually its the argument of the trig function)

Answer 18

would it be 1?

Answer 19

how about if we try 1-t

Answer 20

how?

Answer 21

because its easier to do sin^-1(u) than it is to do sin^-1(u-t)
the u is meant to simplify the equation
do you see why we picked 1-t?

Answer 22

oh okay I see

Answer 23

what would be the derivative of u=1-t?
(hint: its derivative of u with respect to t
notationwise, its u'

Answer 24

is it -1?

Answer 25

yes :)

Answer 26

the next step is to find the derivative of inverse sine
the formula for d/du sin^-1(u) is 1/sqrt(1-u^2)
its similar to inverse cosine, but its positive instead

to solve, we take the derivative of the inverse sine, multiply by the derivative of u, then plug in 1-t back in. what do you get?

Answer 27

is is 1/sqrt (1-1^2)?