A few implicit differentiation problems

[number_13]

1.Use implicit differentiation to find the equation of the tangent line to the curve
xy^3+xy=16 at the point (8,1) . The equation of this tangent line can be written in the form y=mx+b

2.For the equation given below, evaluate y^1 at the point(1,-1) .
(6x-y)^4+2y^3=2399.

3.Find the equation of the tangent line to the curve (a lemniscate)
2(x^2+y^2)^2=25(x^2-y^2) at the point (3,-1). The equation of this tangent line can be written in the form y=mx+b

4.Find the slope of the tangent line to the curve sqrt(3x+3y)+sqrt(xy)=6.48 at the point
(1,5).

5.Use implicit differentiation to find the slope of the tangent line to the curve
y/(x+4y)=x^7+2 at the point (1,-3/11).

Anyone who can help... it would be greatly appreciated!

 

[rocomath]

attempt? you have to show work in order to receive help!

 

[tacman]

1. To find the equation of the tangent line, we first need to find the derivative of the curve. Using implicit differentiation, we have:

d/dx (xy^3+xy) = d/dx (16)
y^3 + 3xy^2 * dy/dx + y + x * dy/dx = 0
dy/dx = - (y^3 + y)/(3xy^2 + x)

Plugging in the point (8,1), we have dy/dx = - (1^3 + 1)/(3*8*1^2 + 8) = -2/25. This is the slope of the tangent line. To find the y-intercept, we plug in the point (8,1) into the equation y = mx + b and solve for b.

1 = (-2/25)*8 + b
b = 211/25

Therefore, the equation of the tangent line is y = (-2/25)x + 211/25.

2. To evaluate y' at the point (1,-1), we first find the derivative of the curve using implicit differentiation:

d/dx [(6x-y)^4+2y^3] = d/dx (2399)
4(6x-y)^3 * (6 - dy/dx) + 3*2y^2 * dy/dx = 0
dy/dx = (4(6x-y)^3 * 6 + 6y^2)/(4(6x-y)^3 + 3*2y^2)

Plugging in the point (1,-1), we have dy/dx = (4(6*1-(-1))^3 * 6 + 6(-1)^2)/(4(6*1-(-1))^3 + 3*2(-1)^2) = 3/4. This is the slope of the tangent line.

3. To find the equation of the tangent line, we first find the derivative of the curve using implicit differentiation:

d/dx [2(x^2+y^2)^2 - 25(x^2-y^2)] = d/dx (0)
4(x^2+y^2) * (2x + 2y * dy/dx) - 50x + 50y * dy/dx = 0
dy/dx = (50x