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Since composing inverse functions gets us back where we started,
sin(arcsin x) = x.
Taking the derivative of each side of this equation gives us
cos(arcsin x) · (arcsin x)' = 1
This is correct, but we're not done. cos(arcsin x) is messy, and we can simplify it.
On the right-hand side, we have (x)' = 1. On the left-hand side, we have an outside function sin(□) and an inside function arcsin x. We need to use the chain rule. The derivative of sin(□) is cos(□), therefore (sin(arcsin x))' = cos(arcsin x) · (arcsin x)'.
The simplification is the part that relies on SOHCAHTOA and the Pythagorean theorem. arcsin x is an angle. Call that angle θ. Then θ = arcsin x means sin θ = x. Since , we know
A triangle representing this equation would look something like this:
Using the Pythagorean Theorem we can find the missing side of the triangle (in terms of x):
We want to know cos(arcsin x) so we can simplify the equation
If arcsin x = θ, then
cos (arcsin x) = cos (θ).
Looking at the triangle and using SOHCAHTOA, we see that