Implicit Differentiation

3.5

Explicit vs. Implicit Functions

• Explicit functions are functions where one variable is clearly expressed as a function of another such as or

• Implicit functions are only implied by an equation, and may be difficult to express explicitly such as

Explicit Vs. Implicit cont.Explicit Implicit

Using product rule on the left side and and then using chain rule (since y is still a function of x)

Since ,

Differentiating with respect to x

•

•

• [x + 3y]

•

Variables agree Use power rule

Variables disagree Use chain rule

Use chain rule

Product RuleChain RuleSimplify

Implicit Differentiation Steps

1) Differentiate both sides of the equation with respect to x.

2) Collect all terms involving on the left side of the equation and move all other terms to the right side of the equation.

3) Factor out of the left side of the equation.4) Solve for by dividing both sides of the equation

by the factor on the left that does not contain

• Most implicit functions can not be defined explicitly.

• If they can be defined explicitly, most of the time you need to restrict the domain.

• Ex. , the implicit equation of the unit circle defines y as a function of x only, if -1 ≤ x ≤ 1 and one considers only non-negative (or non-positive) values for the values of the function.

Find the equation of the line tangent to the circle.

Find ,given that:

• 1) Differentiate both sides with respect to x.• 2) Collect the dy/dx terms on the left side of

the equation.• 3) Factor dy/dx out of the left side of the

equation.• 4) Solve for dy/dx by dividing by (

Implicit Curve represented by 𝑦 3+𝑦 2−5 𝑦−𝑥2=−4

Graphs of differentiable functions

• Let’s represent each of these equations as differentiable functions that we can graph (if possible)

• A) • B) • C) • D)

Determine the slope of the tangent line to the graph

• Ellipse: at point (,-)

• Lemniscate: at (3,1)

• Find the derivative of Inverse Sinusoidal Curve: sin y = x

Finding a line tangent to a graph

• at point ( )

Find the Second Derivative

𝑥2+ 𝑦2=25

Hw

• 1-19 odd, 25, 27, 29, 35-38, 43-46