Find the slope of the graph of the relation x^2y + 4y = 8 at the point (2, 1). 1 3/ 2 1/ 2 -1/2 find y' if x = cos(y). −sin^2(y) sec^2(y) −tan^2(y) −csc(y) if 2x^2 + y^2 = −2 then evaluate d^2y/dx^2 when x = 2 and y = 3. round your answer to 2 decimal places. find dy/dx if f(x) = (x + 1)^2x.

1. 1/2
Find the slope of the graph of the relation x^2y + 4y = 8 at the point (2, 1).
In exponent graph, the slope will be constantly changed so you need to determine a fixed point. The slope in point 2,1 would be:
m= y/x= 1/2

2.  -csc(y)
Find y' if x = cos(y)

x = cos(y)
dx = -sin(y)dy
dy/dx = -1/sin(y)
dy/dx = -csc(y)
y' = -csc(y)

3.  y''= -0.48
If 2x^2 + y^2 = −2 then evaluate d^2y/dx^2 when x = 2 and y = 3. Round your answer to 2 decimal places.

First you need differentiate the equation implicitly
2x^2 + y^2 = −2
dy/dx=  2*2 x^(2-1) + 2*y^2-18y'
4x+4y'y=0
4y'y=-4x
y'= -x/y

Then find out the y'' using quotient differentiation
y'= -x/y
y''=(−y + xy')/y² 

If x=2 and y=3 then
y'= -x/y
y'= -2/3

y''=(−y + xy')/y² 
y''=(−3 + 2*(-2/3))/3² 
y''=(−9/3 - 4/3)/9 
y''=(-13/3)/9 = -13/27
y''= -0.48

4. answer = 2( x+1) ^(2)
Find dy/dx if f(x) = (x + 1)^2x.
In this question, the number is (x+1) power 2x. You don't need to touch the (x+1). Just do the differentiation focusing on the 2x. 
y= (x + 1)^2x
dx/dy= 2*( x+1)^(1*2 x^1-1)
dx/dy= 2( x+1) ^(2)