Question

Find dy/dx for x2 + y2 = 2xy.

2. If 5x2 + y4 = -9 then evaluate d^2y/dx^2 when x = 2 and y = 1. Round your answer to two decimal places. Use the hyphen symbol, -, for negative values.

3. find dy/dx if f(x)=(x+1)^2x

4. Find the slope of the graph of the relation x2y + 4y = 8 at the point (2, 1).

Answer 1

x^2 + y^2 = 2 x y
2 x + 2 y dy/dx = 2 x dy/dx + 2 y

dy/dx (y-x) = y-x

dy/dx = 1
easier way without using calculus:
x^2-2xy+y^2 = 0
(x-y)(x-y) = 0
x = y
slope = 1

Answer 2

f(x)=(x+1)^2x

d/dx u^v = v u^(v-1)du/dx + u^v lnu dv/dx
so
f'=2x(x+1)^(2x-1) + 2(x+1)^2x ln(x+1)

Answer 3

Find the slope of the graph of the relation x2y + 4y = 8 at the point (2, 1)

(x^2+4)y = 8

(x^2+4)dy/dx + y (2x) = 0

8 dy/dx + 4 = 0
dy/dx = -1/2

Answer 4

If 5x2 + y4 = -9 then evaluate d^2y/dx^2 when x = 2 and y = 1

5 x^2 + y^4 = -9
10 x + 4 y^3 dy/dx = 0
so
dy/dx = -2.5 x/y^3
d^2y/dx^2=-2.5[ y^3 - 3 x y^2 dy/dx]/y^6
at (2,1)
dy/dx = -2.5(2/1) = -5
so
d^2y/dx^2=-2.5[1-6(-5)]/1
= -2.5 [ 31]
= - 77.5