Implicit Differentiation 3.7

Explicit vs Implicit Functions y = 5 – 2x 2x + y = 5 y2 – x2 = 4 xy = 12 What’s the difference?

Using the Chain Rule - Implicitly Suppose y is a function of x. Then and

Voting Question Suppose y is a function of x. A) 1 B) C) D) x + y

Explicit vs Implicit differentiation Example: Find if x3 + y3 = 1

Implicit Differentiation Given an implicit function, to find differentiate each term of the equation with respect to x. When differentiating a y-term, remember to multiply the derivative by (using the chain rule). Collect all terms on one side of the equation. Then factor out and divide both sides by its coefficient to solve for .

Voting Question – Step 1 The problem: Find if 5x3 + 4y2 = 8y Step 1: Differentiate each term with respect to x. After this step, we have: A. B. C. D.

Voting Question – Step 2 The problem: Find if 5x3 + 4y2 = 8y Step 2: Collect all terms on one side of the equation. After this step we have, A. B. C. D.

Voting Question – Step 3 Step 3: Solve for . After this step, we have: The problem: Find if 5x3 + 4y2 = 8y Step 3: Solve for . After this step, we have: A. B. C. D.

More Examples –Implicit Differentiation Find for each of the following. 3xy – y2 = 1 cos(xy2) = x x2y3 – 6 = 5y3 + x And how about an equation of the tangent line @ (2, -2)?

Related Rates Chapter 3.8

Introducing – Related Rates A typical problem: A balloon in the shape of a sphere is being blown up at a rate of 80 cm3/min. At what rate is the radius of the balloon changing at the instant when the radius is 10 cm?

Setting up the problem What we know: The volume and radius of a sphere are related to one another by the formula . Both the radius and volume are changing as the balloon is being blown up. The rates at which radius and volume change with time are themselves related. An equation showing how these rates of change are related can be obtained by differentiating implicitly with respect to time.

Solving the Problem Step 1: Write the equation that relates volume and radius: Step 2: Differentiate this equation implicitly with respect to time: Step 3: Plug in the given values and solve for the unknown.

Completing the solution We have the related rates equation . The problem says: A balloon in the shape of a sphere is being blown up at a rate of 80 cm3/min. At what rate is the radius of the balloon changing at the instant when the radius is 10 cm? We substitute = 80 and r = 10 into the related rates equation to find cm/min.

Another Example A stone dropped into water creates circular ripples of increasing size. If the radius increases at a constant rate of 0.12 cm/sec, at what rate is the area changing at the instant that the radius is 3 cm?

Setting up the Problem Write the formula that shows the relationship between area and radius of a circle. A. A = 2 π r B. A = π r2 C. A = 2 π r2 D. A = ½ π r

Differentiating with respect to time Taking the derivative of both sides of the equation with respect to time gives: A. B. C. D.

Plugging in the Given Values The problem says: If the radius increases at a constant rate of 0.12 cm/sec, at what rate is the area changing at the instant that the radius is 3 cm? Substituting the given values into the related rates equation gives: A. .12 = 2π(3) B. = 2π(3)(.12) C. = π(32)(.12) D. .12 = π(32)

New problem: The sliding ladder Joey is perched on top of a 10-foot ladder leaning against the back wall of an apartment building (spying on a neighbor) when it starts to slide down the wall at a rate of 4 ft per minute. Joey's accomplice, Lou, is standing on the ground 6 ft. away from the wall. How fast is the base of the ladder moving when it hits Lou?

One more: The growing sandpile Sand falls from an overhead bin and accumulates in a pile in the shape of a cone with a radius that is always 2 times its height. If sand is dumped onto the pile at a rate of 6 cm3/sec, at what rate is the height of the pile increasing when it is 10 cm high?