AP Calculus ABChapter 2, Section 5Implicit Differentiation2013 - 2014

Implicit vs. Explicit• Up to this point you have been working with _____________

derivatives.

• Example: you can find the derivative of y with respect to x of this function…

Implicit vs. Explicit• Some functions are only implied by an equation.

• For instance, the function is defined implicitly by the equation .

Implicit Form Explicit Form Derivative

Implicit vs. Explicit• What happens when you get an equation like this?

• This is when you must use ___________ differentiation.

• Implicit differentiation means you can still differentiation x like always, but when you differentiation y values, you must apply the chain rule because you are assuming that y is defined implicity as a differentiable function of x.

Differentiation with Respect to x

𝑑𝑑𝑥

[𝑥3]

Differentiation with Respect to x

𝑑𝑑𝑥

[𝑦3]

Differentiation with Respect to x

𝑑𝑑𝑥

[𝑥+3 𝑦 ]

Differentiation with Respect to x

𝑑𝑑𝑥

[𝑥 𝑦2]

Guidelines for Implicit Differentiation

1. Differentiate both sides of the equation with respect to x2. 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 .

Implicit Differentiation• Find given that

Implicit Differentiation• It is meaningless to solve for in an equation that has no

solution points.• For example: has no solution points• If a segment of a graph can be represented by a differentiable

function, will have meaning as the slope at each point on the segment.

Representing a Graph by Differentiable Functions

• If possible, represent y as a differentiable function of x.

Representing a Graph by Differentiable Functions

• If possible, represent y as a differentiable function of x.

Representing a Graph by Differentiable Functions

• If possible, represent y as a differentiable function of x.

Finding the Slope of a Graph Implicitly

• Determine the slope of the tangent line to the graph of at the point

Finding the Slope of a Graph Implicitly

• Determine the slope of the graph of at the point (3, 1).

Determining a Differentiable Function• Find implicitly for the equation . Then find the largest

interval of the form on which y is a differentiable function of x.

Finding the Second Derivative Implicitly

• Given , find

Finding a Tangent Line to a Graph

• Find the tangent line to the graph given by at the point (

Ch. 2.5 Homework• Pg. 146 – 147, #’s: 7, 15, 17, 23, 27, 29, 33, 41, 49, 57, 65

• 11 Total problems