Chapter : Derivatives Section 3.7: Implicit Differentiation AP CALCULUS AB Chapter : Derivatives Section 3.7: Implicit Differentiation

What you’ll learn about Implicitly Defined Functions Lenses, Tangents, and Normal Lines Derivatives of Higher Order Rational Powers of Differentiable Functions … and why Implicit differentiation allows us to find derivatives of functions that are not defined or written explicitly as a function of a single variable.

Implicitly Defined Functions

Section 3.7 – Implicit Differentiation Explicit Form of an equation – is expressed in the form Ex: Implicit Form of an equation – when both x and y are used throughout the equation

Implicitly Defined Functions

Example Implicitly Defined Functions

Implicit Differentiation Process

Section 3.7 – Implicit Differentiation Guidelines for Implicit Differentiation: 1. Differentiate both sides of the equation with respect to x. (Treat y as a function of 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 the right side of the equation by the part of the left side of the equation without the

Section 3.7 – Implicit Differentiation Example:

Section 3.7 – Implicit Differentiation Another example:

Lenses, Tangents and Normal Lines In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light makes with the line perpendicular to the surface of the lens at the point of entry (angles A and B in Figure 3.50). This line is called the normal to the surface at the point of entry. In a profile view of a lens, the normal is a line perpendicular to the tangent to the profile curve at the point of entry. Implicit differentiation is often used to find the tangents and normals of lenses described as quadratic curves.

Lenses, Tangents and Normal Lines

Example Lenses, Tangents and Normal Lines

Example Lenses, Tangents and Normal Lines

Example Derivatives of a Higher Order

Section 3.7 – Implicit Differentiation Higher Order Derivatives:

Section 3.7 – Implicit Differentiation

Rule 9 Power Rule For Rational Powers of x