Implicit Diffentiation

Question

Use implicit differentiation to find the slope of the tangent line to the curve

y/(x-4y)= x^2 +8


at point (1, 9/37)

Answer 1

Let multiply (x - 4y) on both sides of equation to get rid of denominator.

 

y = (x - 4y)(x² + 8)

 

y = x³ - 4x²y + 8x - 32y

 

Now we can differentiate both sides of equation using the chain rule.

 

y' = 3x² - [8xy + 4x²y'] + 8 - 32y'

 

y' = 3x² - 8xy - 4x²y' + 8 - 32y'

 

Move all the y' terms to one sides of equation and all the non-y' terms to the other side of equation.

 

y' + 4x²y' + 32y' = 3x² - 8xy + 8

 

Factor out the y'.

 

y' (1 + 4x² + 32) = 3x² - 8xy + 8

 

Divide both sides of equation by (1 + 4x² + 32).

 

y' = (3x² - 8xy + 8) / (1 + 4x² + 32)

 

Substitute the values of the point (1, 9/37) into y'.

 

y' = [3(1)² - 8(1)(9/37) + 8] / [1 + 4(1)² + 32]

 

y' = (335/37) / (37)

 

y' = 335/1369