If y=e^2sinx, then d^2y/dx^2 = ?

Question

Finding the second derivative in terms of x for e^2dinx.

Answer 1

use chain rule to find derivative

let U= 2 sin(x). Then du/dx = 2 coa(x)

then y = e^u. So dy/du = e^u

and therefore, by cahin rule: dy/dx = dy/du * du/dx

Dy/dx = e^2sin(x)(2cos(x))= 2 cos(x) e^{2 sin(x)}

use product rule on dy/dx: (uv)’ = uv’ + vu’, where

u= 2 cos(x) & v= e^2sin(x)

hence

d²/dx² = 2 cos(x) { 2 cos(x) e^{2 sin(x)} } - 2 sin(x) e^2sin(x)

= 2 e^{2 sin(x)} {2cos²(x) - sin(x)

Answer 2

y = e^2sin(x)

dy/dx = 2cos(x)e^2sin(x)

d²y/dx² = 2[-sin(x)e^2sin(x) + cos(x)(2cos(x))e^2sin(x)]

= 4cos²(x)e^2sin(x) - 2sin(x)e^2sin(x)