Question

Calculus 3.10 Linearization (a) y=x^2sin2x (b) y=ln sqrt(1+t^2) find the differential of each.

Answer 1

Below in bold.

Explanation:

(a) y=x^2sin2x

Using the Product Rule:

dy/dx = x^2*2cos2x + 2xsin2x

dy/dx = 2x(xcos2x + sin2x).

(b) y=ln sqrt(1+t^2)

y = ln (1 + t^2)^1/2

Using the chain rule:

dy/dx = 1 / (1 + t^2)^1/2 * 1/2(1 + t^2)^-1/2 * 2t

= 2t/(1 + t^2)^1/2 * 1/ (2(1 + t^2)^1/2

= 2t / 2(1 + t^2)

= t / ( 1 + t^2).