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2.5 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

Write y explicitly as a function of x and take the derivative. Calculus Warm-up before 2.5

Distinguish between functions written in implicit form and explicit form. Use implicit differentiation to find the derivative of a function. Objectives

Implicit and Explicit Functions

Most functions have been expressed in explicit form. For example, in the equation the variable y is explicitly written as a function of x. Some functions, however, are only implied by an equation. For instance, the function y = 1/x is defined implicitly by the equation xy = 1. Explicit form

Suppose you were asked to find dy/dx for this equation. You could begin by writing y explicitly as a function of x and then differentiating. This strategy works whenever you can solve for y and write it explicitly as a function of x. You cannot, however, use this procedure when you are unable to solve for y as a function of x. Implicit and Explicit Functions

For instance, how would you find dy/dx for the equation where it is very difficult to express y as a function of x explicitly? To do this, you can use implicit differentiation. Implicit and Explicit Functions

Use implicit differentiation when it is impossible or impractical to express y as a function of x explicitly. To find dy/dx implicitly, realize that differentiation is taking place with respect to x. This means that when you differentiate terms involving x alone, you can differentiate as usual. However, when you differentiate terms involving y, you must apply the Chain Rule, because you are assuming that y is defined implicitly as a differentiable function of x.

Example 1 – Differentiating with Respect to x

Example 1 – Differentiating with Respect to x, cont’d

Its not necessary to solve for y because we can take this derivative implicitly. Do the same thing to both sides. Note use of chain rule.

This can’t be solved for y. This technique is called implicit differentiation. 1 Differentiate both sides w.r.t. x. 2 Solve for.

You try:

We need the slope. Since we can’t solve for y, we use implicit differentiation to solve for. Find the equations of the lines tangent and normal to the curve at. Note product rule.

Find the equations of the lines tangent and normal to the curve at. tangent:normal:

Higher Order Derivatives Find if Substitute back into the equation.

Derivative formulas include the chain rule! etcetera… If formulas on a memorization sheet are written with instead of. Don’t forget to include the term!

Homework 2.5 Day 1: pg.146: 1-15 odd, odd, 45, 57 Day 2: MMM 53-54