Which of the ff function cannot be expressed in explicit form of dy/dx
4x+4y=2
xy-y^2=1
ycosx=5
My answer is ycosx=5
clearly #1 can be so expressed.
#2 cannot: y' = y/(2y-x)
#3 can: y = 5secx, so y' = 5secx tanx
To determine which of the given functions cannot be expressed in explicit form of dy/dx, we need to determine if it is possible to solve each equation for y in terms of x and then differentiate it to find dy/dx.
Let's go through each function:
1. 4x + 4y = 2:
To solve for y, we need to isolate it on one side of the equation. Then, we can differentiate the equation to find dy/dx.
4y = 2 - 4x
y = (2 - 4x) / 4
Differentiating y with respect to x, we get:
dy/dx = -4/4 = -1
Therefore, this function can be expressed in explicit form of dy/dx.
2. xy - y^2 = 1:
To solve for y, we need to isolate it on one side of the equation and differentiate the equation to find dy/dx.
xy = y^2 + 1
y^2 - xy + 1 = 0
This is a quadratic equation in terms of y. When we rearrange the equation for y, we'll have two possible values for y for a given x, resulting in dy/dx being multivalued. Therefore, this function cannot be expressed in explicit form of dy/dx.
3. ycosx = 5:
To solve for y, we divide both sides of the equation by cos(x) and differentiate the equation to find dy/dx.
y = 5/cos(x)
Differentiating y with respect to x, we get:
dy/dx = 5 * (-sin(x)) / cos^2(x) = -5sin(x)/cos^2(x) = -5tan(x)
Therefore, this function can be expressed in explicit form of dy/dx.
In conclusion, the function that cannot be expressed in explicit form of dy/dx is xy - y^2 = 1.