Implicit Differentiation

Implicitly vs. Explicitly Defined Functions

y is given explicitly as a function of x (y is solved in terms of x)

y is given implicitly as a function of x (y is NOT solved in terms of x)

Some Graphs of Implicitly Defined Expressions

Graph of x-2xy=2

Graph of x2+y2=1

Graph of cos(x2)-sin(y2)=0

Implicit Differentiation

When y is not given explicitly as a function of x, think of y as “locally” being a function of x.

This approach is used to differentiate expressions where a term in the expression is considered implicitly as a function of another variable.

Graph of x-2xy=2

Find the equation of the tangent line at the point (2,0)

Graph of x2+y2=1

Find the equation of the tangent line at the point (√0.5, √0.5)