Implicit Differentiation Problems and Related Rates

Use implicit differentiation to find the derivative of the following: How can we measure the rate of change for a non-linear function? December 6, 2013 Take out: pencil, hw, notebook Homework: Page 251 problems 1,2,9,11,14 (remember the odd problems answers are in the back use that to help you not do your HW) Objectives: You will be able to… Use implicit differentiation to find the derivative of an inseparable function Use implicit differentiation to solve related rates problems. Do Now: Use implicit differentiation to find the derivative of the following: Announcement: Related Rates/optimization quiz Monday Agenda: Do Now HW Questions Multiple variable implicit differentiation Related rates problems Homework

Do Now Use implicit differentiation to find the derivative of the following:

Review Find the derivative of y with respect to x of…

What if you have more than 2 variables? Whenever you take the derivative of something that isn’t the variable your differentiating with respect to, then you multiply by the derivative Example:

You can even differentiate by a variable that isn’t there! For example: Differentiate the following with respect to time, t.

Differentiate the following with respect to time, t.

Related Rates Problems Some problems can best be solved using the implicit differentiation method we just learned. For these problems, the trick is to follow the method!

Related Rates Method: Use this every time! Read the problem carefully, underline/highlight key information/words. Draw a detailed diagram with labels Write down the variables and rates that you know / don’t know Write the equation relating the variables with constants plugged in (you will not always have a constant to plug in this is only when you need to get rid of a rate that you are not trying to solve for or given) Differentiate (usually with respect to t) and attempt to solve for the missing variable. If not all of the needed information is available, pause. Go back to the original equation (or some other relationship) to find the missing variable.

Example 2: Water comes out of faucet at a rate of 5 cm3/sec Example 2: Water comes out of faucet at a rate of 5 cm3/sec. The water flows into a glass with radius 3 cm. What is the rate at which the height is increasing? 5 cm3/s

A 5 meter ladder is leaned against a building A 5 meter ladder is leaned against a building. The base of the ladder is pulled away from the building at 3 m/s. How fast will the top of the ladder be moving when the base of the ladder is 4 meters from the building? ?m/s (constant) c = 5m b 3 m/s a=4 m

You Try!: A rocket that is launched vertically is tracked by a radar station located on the ground 3 miles from the launch site. What is the vertical speed of the rocket at the instant that its distance from the radar station is 5 miles and this distance is increasing at a rate of 5000 mph? c=5 miles b Launch Pad a=3 miles