i don't know which one is match. i tried those problem. i couldn't solve it

Find y' by implicit differentiation. Match the expressions defining y implicitly with the letters labeling the expressions for y'.
1. 7xsiny+2cos2y=2cosy
2. 7xcosy+2sin2y=2siny

a. 7cosy/7xsiny-4cos2y+2cosy

b. -(7siny/7xcos-4cos2y+2siny)

c. 7siny/4sin2y-xcosy-2siny

d. 7cosy/7xsiny+4sin2y+2cosy

1.

7xsiny+2cos2y=2cosy
7siny + 7xcosy*y' - 4sin2y*y' = -2siny*y'

y' = 7siny/(-7xcosy + 4sin2y - 2siny)
is there a typo in (b) or (c)?

2.
7xcosy+2sin2y=2siny
7cosy - 7xsiny*y' + 4cos2y*y' = 2cosy y'
y' = 7cosy/(2cosy + 7xsiny - 4cos2y)
= (a)

To find y' by implicit differentiation, we need to differentiate both sides of the equation with respect to x.

Let's start with equation 1: 7xsiny+2cos2y=2cosy

Taking the derivative of both sides with respect to x:

d/dx(7xsiny) + d/dx(2cos2y) = d/dx(2cosy)

Using the product rule and chain rule, we can differentiate the terms on the left side:

(7siny + 7xcosy * dy/dx) + (-4sin2y * dy/dx) = -2siny * dy/dx

Now we can solve for dy/dx by isolating the terms with dy/dx:

7siny - 4sin2y * dy/dx + 7xcosy * dy/dx = -2siny * dy/dx

Moving all the terms with dy/dx to one side:

(7xcosy - 4sin2y + 2siny) * dy/dx = -7siny

Now we can solve for dy/dx by dividing both sides by (7xcosy - 4sin2y + 2siny):

dy/dx = -7siny / (7xcosy - 4sin2y + 2siny)

So for equation 1, the expression defining y' is:

a. 7cosy/7xsiny-4cos2y+2cosy

Now let's move on to equation 2: 7xcosy+2sin2y=2siny

Following the same steps as above, we can differentiate both sides with respect to x:

d/dx(7xcosy) + d/dx(2sin2y) = d/dx(2siny)

By applying the product rule and chain rule:

(7cosy * dy/dx + 7x * -siny * dy/dx) + (4cos2y * dy/dx) = 2cosy * dy/dx

Grouping the terms with dy/dx:

(7cosy - 4cos2y + 2siny * y) * dy/dx = -7x * siny

Dividing both sides by (7cosy - 4cos2y + 2siny):

dy/dx = -7x * siny / (7cosy - 4cos2y + 2siny)

For equation 2, the expression defining y' is:

b. -(7siny/7xcos-4cos2y+2siny)

In conclusion, the expressions for y' are:

1. a. 7cosy/7xsiny-4cos2y+2cosy
2. b. -(7siny/7xcos-4cos2y+2siny)

To find y' by implicit differentiation in this problem, you need to take the derivative of the entire equation with respect to x and then solve for y'.

Let's go through the steps for each expression and match them with the letters labeling the expressions for y'.

1. Start by differentiating the equation with respect to x:
- For the left side, use the product rule and chain rule:
- The derivative of 7xsiny with respect to x is 7siny + 7xcosy * (dy/dx)
- The derivative of 2cos2y with respect to x is -4sin2y * (dy/dx)
- For the right side, since it's a constant, its derivative is zero.
Therefore, the derivative of the entire equation becomes:
7siny + 7xcosy * (dy/dx) - 4sin2y * (dy/dx) = 0

Comparing this expression with the options:

a. 7cosy / 7xsiny - 4cos2y + 2cosy
This expression contains the terms 7cosy / 7xsiny and -4cos2y, but it does not have the term 7siny, which appears in the derivative.

b. -(7siny / 7xcosy - 4cos2y + 2siny)
This expression has the correct term -(7siny / 7xcosy), but it does not have the term 7xcosy.

c. 7siny / 4sin2y - xcosy - 2siny
This expression contains the terms 7siny / 4sin2y and -2siny, but it does not have the term 7xcosy.

d. 7cosy / 7xsiny + 4sin2y + 2cosy
This expression contains the correct terms 7cosy / 7xsiny, 4sin2y, and 2cosy.

So, the correct match for the expression defining y' in the first problem is d.

Let's proceed to the second problem:

2. Differentiate the equation with respect to x:
- For the left side, use the product rule and chain rule:
- The derivative of 7xcosy with respect to x is 7cosy + 7xcos2y * (dy/dx)
- The derivative of 2sin2y with respect to x is 4cos2y * (dy/dx)
- For the right side, since it's a constant, its derivative is zero.
Therefore, the derivative of the entire equation becomes:
7cosy + 7xcos2y * (dy/dx) + 4cos2y * (dy/dx) = 0

Comparing this expression with the options:

a. 7cosy / 7xsiny - 4cos2y + 2cosy
This expression contains the correct terms 7cosy and 2cosy, but it does not have the term 7xcos2y.

b. -(7siny / 7xcosy - 4cos2y + 2siny)
This expression has the correct term -(7siny / 7xcosy), but it does not have the terms 7cosy or 4cos2y.

c. 7siny / 4sin2y - xcosy - 2siny
This expression does not have any of the correct terms - 7cosy, 7xcos2y, or 4cos2y.

d. 7cosy / 7xsiny + 4sin2y + 2cosy
This expression does not contain the terms 7xcos2y or 4cos2y.

Unfortunately, none of the provided options match the expression defining y' for the second problem.

My suggestion would be to revisit your calculations or consider alternative methods to find y'.