Question

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.

Answer 1

Answer: The correct answer is -1/2

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Answer 2

1. Answer: 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. Answer: -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. Answer: 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)