What is the derivative of sin(arc cosx)?

1 Answer
Jim H
Jun 12, 2015

-x/sqrt(1-x^2)

Explanation:

y=sin(arccosx)

Method 1
(Use if you don't remember the derivative of arccosx., but remember your trigonometry.)

sin(arccosx) is the sine of a number (an angle) in [0, pi] whose cosine is x.
In [0, pi] the sine is positive, so: sin(arccosx) = sqrt(1-x^2)

Now y=sin(arccosx) = sqrt(1-x^2), so

y' = 1/(2sqrt(1-x^2)) (-2x) = -x/sqrt(1-x^2)

Method 2:
(Use if you remember the derivative of arccosx.)
Use the chain rule. (And the fact that cos(arccosx)=x)

y'=cos(arccosx) * d/dx(arccosx)

= x* (-1/sqrt(1-x^2)) = -x/sqrt(1-x^2)

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