Find the slope of the tangent line

Question

Find dy/dx for x²y⁴ + √2x + 5y=7

Show that the point (x,y) = (2,1) lies on the curve defined by the equation above and find the slope of the tangent line at this point. 

Answer 1

Use implicit differentiation using the chain rule to find dy/dx.  Your derivative will be in terms of x and y.  Plug in x=2 and x=1 to evaluate dy/dx.  This will give you the slope of the tangent line.

 

Answer 2

x²y⁴ + (2x + 5y)^½ = 7

 

Plugging in x = 2 and y = 1, we get 7 = 7.  So, the point (2,1) lies on the curve.  

 

Implicitly differentiate the given function to obtain:

 

2xy⁴ + 4y³(dy/dx)(x²) + (½)(2x + 5y)^-½(2 +5dy/dx) = 0

 

Plug in x = 2 and y = 1:

 

  4 + 16(dy/dx) + (1/6)(2 + 5dy/dx) = 0

 

  4 + 16(dy/dx) + 1/3 + (5/6)(dy/dx) = 0

 

                                   (101/6)(dy/dx) = -13/3

 

                                                dy/dx = (-13/3)(6/101) = -26/101