- #1
QuarkCharmer
- 1,051
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Homework Statement
Prove that:
[tex]\frac{d}{dx}arcsin(x) = \frac{1}{\sqrt{1-x^{2}}}[/tex]
Homework Equations
The Attempt at a Solution
[tex]y=arcsin(x)[/tex]
[tex]sin(y)=sin(arcsin(x))[/tex]
[tex]sin(y)=x[/tex]
[tex]\frac{d}{dx}(sin(y)=x)[/tex]
[tex]cos(y)\frac{dy}{dx}=1[/tex]
[tex]\frac{dy}{dx}=\frac{1}{cos(y)}[/tex]
Using a triangle:
[tex]cos(y)=\sqrt{1-x^2}[/tex]
[tex]\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}[/tex]
[tex]\frac{d}{dx}arcsin(x) = \frac{1}{\sqrt{1-x^{2}}}[/tex]
We were shown these in class before starting trig-sub and all that, and the professor said that there are proofs by differentiation. I don't know how to explain the steps where I determine that the cosine of the angle y is equal to sqrt(1-x^2) without drawing out a triangle. Is there some way? Other than that I think I have it figured out and I can do the same for the other functions.