Section 2.5 Day 1 AP Calculus

Learning Targets Define implicit and explicit functions Evaluate derivatives using Implicit Differentiation

Explicit Form Everything is solved for one variable Ex: 𝑦=3𝑥+1

Implicit Form There is a mix of variables on the same side Ex: 𝑥𝑦=1

Differentiating with Respect to x Ex: 𝑑 𝑑𝑥 ( 𝑥 3 ) = 3 𝑥 2 𝑑𝑥 𝑑𝑥 = 3 𝑥 2 Ex: 𝑑 𝑑𝑥 ( 𝑦 3 ) = 3 𝑦 2 𝑑𝑦 𝑑𝑥

Implicit Differentiation Steps 1. Take the derivative of each piece 2. Put 𝑑𝑦 𝑑𝑥 after the derivative of “y” terms 3. Split 𝑑𝑦 𝑑𝑥 terms away from other terms using the equal sign 4. Factor 𝑑𝑦 𝑑𝑥 5. Divide

Example 1 𝑑 𝑑𝑥 [𝑥 𝑦 2 ] 𝑥 2𝑦 𝑑𝑦 𝑑𝑥 + 𝑦 2 2𝑥𝑦 𝑑 𝑑𝑥 + 𝑦 2

Example 2 Find the derivative of 𝑦 3 − 𝑦 2 −5𝑦 − 𝑥 2 =−4 1. 3 𝑦 2 𝑑𝑦 𝑑𝑥 −2𝑦 𝑑𝑦 𝑑𝑥 −5 𝑑𝑦 𝑑𝑥 −2𝑥=0 2. 3 𝑦 2 𝑑𝑦 𝑑𝑥 −2𝑦 𝑑𝑦 𝑑𝑥 −5 𝑑𝑦 𝑑𝑥 =2𝑥 3. 𝑑𝑦 𝑑𝑥 3 𝑦 2 −2𝑦−5 =2𝑥 4. 𝑑𝑦 𝑑𝑥 = 2𝑥 3 𝑦 2 −2𝑦−5

Example 3 Find the derivative of ( 𝑥 2 +4 𝑦 2 =4) 1. 2𝑥+8𝑦 𝑑𝑦 𝑑𝑥 =0 2. 8𝑦 𝑑𝑦 𝑑𝑥 =−2𝑥 3. 𝑑𝑦 𝑑𝑥 =− 2𝑥 8𝑦 =− 𝑥 4𝑦

Example 4 𝑑𝑦 𝑑𝑥 3 𝑥 2 + 𝑦 2 2 =100𝑦 1. 6 𝑥 2 + 𝑦 2 2𝑥+2𝑦 𝑑𝑦 𝑑𝑥 =100 𝑑𝑦 𝑑𝑥 2. 6 𝑥 2 +6 𝑦 2 2𝑥+2𝑦 𝑑𝑦 𝑑𝑥 =100 𝑑𝑦 𝑑𝑥 3. 12 𝑥 3 +12 𝑥 2 𝑦 𝑑𝑦 𝑑𝑥 +12 𝑦 2 𝑥+12 𝑦 3 𝑑𝑦 𝑑𝑥 =100 𝑑𝑦 𝑑𝑥 4. 12 𝑥 2 𝑦 𝑑𝑦 𝑑𝑥 +12 𝑦 3 𝑑𝑦 𝑑𝑥 −100 𝑑𝑦 𝑑𝑥 =−12 𝑥 3 −12 𝑦 2 𝑥 5. 𝑑𝑦 𝑑𝑥 = −12 𝑥 3 −12 𝑦 2 𝑥 12 𝑥 2 𝑦+12 𝑦 3 −100

Example 5 What is the slope of the line tangent to the curve 2 𝑥 2 −3 𝑦 2 =2𝑥𝑦−6 at the point (3, 2)?

Example 6 Find 𝑑 2 𝑦 𝑑 𝑥 2 (2 𝑥 3 −3 𝑦 2 =8)

Exit Ticket for Feedback Find the derivative of 2𝑥+𝑦 2 −𝑥𝑦=10