M2 Differentiation

171. Let z=2^x
ln z = x ln2
Differentiating with respect to x,
(1/z) (dz/dx) = ln 2
dz/dx = z ln2 = (ln2) (2^x)

so, dy/dx = (ln2) (2^x) since 3e^(-3e) is a constant


(b) Let (dx/dy) = u = - (1/24)(3+4x)^2 . [given]

dx^2/dy^2
= d(dx/dy)/dy
= du/dy
= (du/dx) * (dx/dy) [Chain Rule]
= (du/dx) * u
= d [- (1/24)(3+4x)^2]/dx * [- (1/24)(3+4x)^2] [substitution]
= - (1/24) 2(3+4x) (4) * [- (1/24)(3+4x)^2] [Chain Rule]
= (1/72) (3+4x)^3