Calculus: derivative of an inverse: derivative of arcsin
The tutor shows the derivative of arcsin, the inverse of sin.
In yesterday’s post I explained the formula for the derivative of an inverse function
(m-1(x))’ = 1/m'(m-1(x))
Today, I’ll use it to find the derivative of “inverse sin(x)”, aka sin-1(x), aka arcsin(x).
Let’s start with sin(arcsin(x)) = 1, which leads to, from the formula,
arcsin'(x) = 1/cos(arcsin(x))
Now, behold:
We see, in the illustration, that arcsin(x) = θ. The formula becomes
arcsin'(x) = 1/cos(θ)
Once again, from the illustration: cos(θ) =
So we have
arcsin'(x) = 1/
Source:
Larson, Roland E. and Robert P. Hostetler. Calculus, 3rd ed. Toronto: D C Heath and Company, 1989.
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.
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