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 x 2 +y 2 =1

Graph of cos(x 2 )-sin(y 2 )=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 x 2 +y 2 =1 Find the equation of the tangent line at the point (√0.5, √0.5)