Answer to Question #239238 in Calculus for RED

Question

1.Find the second and third derivatives in the simplest form of

y=x⁵-3x²-2x+5

2. Find by implicit differentiation of

z²-3zy+y²=6z-2y

Answer 1

Q1. Given function $y=x^5-3x^2-2x+5\\$

$y'=\frac{dy}{dx}=5x^4-6x-2\\ y''=\frac{d^2y}{dx^2}=20x^3-6x-2\\ y'''=\frac{d^3y}{dx^3}=60x^2-6$

Q2. Given function $z^2-3zy+y^2=6z-2y$

$2z\frac{dy}{dz}-3y\frac{dy}{dz}-3z+2y=6\frac{dy}{dz}-2\\ \Rightarrow (2z-3y-6)\frac{dy}{dz}=3z-2y-2\\ \Rightarrow \frac{dy}{dz}=\frac{(3z-2y-2)}{(2z-3y-6)}$