3.7 Implicit Differentiation

We use implicit differentiation when differentiating both sided of an equation with respect to x and then solving the resulting equation for

• Use when you can’t solve the equation for y
• Differentiate both sides in terms of x
• Solve for y’

Orthogonal – Perpendicular tangents @ points of intersection

Example 1

If x⁴ + y^{4 =} 85 find

1) Differentiate both sides of the equation

x⁴ + y⁴ = 85

2) Remember that y is a function of x and using the chain rule we have,

Therefore,

= = ßFinal answer!

Examples:

Find y’

x³+y³=6xy

d/dx (x³)+d/dx (y³)=d/dx (6xy)

d/dx (x³)+d/dy (y³) dy/dx=d/dx (6xy) -----use product rule

3x²+3y²y’=6xy’+6y

y’(3y²-6x)=6y-3x²

y’=(6y-3x²)/(3y²-6x)=(2y-x²)/(y²-2x)

Find y’

xsiny+cos2y=cosy

xcosyy’+siny-sin2y(2y’)=-sinyy’

xcosyy’-sin2y(2y’)+sinyy’=-siny

y’(xcosy-2sin2y+siny)=-siny

y’=(-sinx)/(xcosy-2sin2y+siny)