Find the slope of the line tangent to y=3xarcsinx at (1/2,pi/4)

Question

Find slope of line tangent to y=3xarcsinx at (1/2,π/4)

Answer 1

In order to find the slope of the tangent line, we need to find the derivative:

y' = (d/dx)(3x arcsin(x))

apply the product rule:

y' = 3 arcsin(x) + 3x (d/dx)(arcsin(x))

the derivative of arcsin(x) can be looked up in a table: (d/dx) (arcsin(x)) = 1/√(1-x²)

If you'd like a proof of this, take a look at this page:

https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-b-implicit-differentiation-and-inverse-functions/session-15-implicit-differentiation-and-inverse-functions/MIT18_01SCF10_Ses15c.pdf

so the derivative is:

y' = 3 arcsin(x) + 3x / √(1-x²)

now just plug in 1/2 (and note that arcsin(1/2) = π/6

y'(1/2) = 3 * (π/6) + 3(1/2) √(1-(1/2)²) which simplifies to π/2 + √3 or ~ 3.302

Answer 2

Hi,

Hopefully this helps you. Let me know if you have any questions.

Cheers,

Darryl.