Find an equation of the tangent line to the graph of the function at the given point.

Question

y = 3 arcsin x (1/2,pi/2)

y=?

Answer 1

Need to find the derivative of y=3sin^-1(x) and use

y-y₀=y'(x-x₀)

There are a couple ways to find y'

x = sin(y/3)

Take the implicit derivative

1 = cos(y/3)/3 y'

y' = 3/cos(y/3)

At (1/2, π/2), y' = 3/(√3 /2) = 6/√3 = 2√3

y - π/2 = 2√3(x-1/2)

y = (2√3)x-√3 + π/2

Note, you can also use (3sin^-1(x))' = 3/√(1-x²) and get the same answer