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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