Chain Rule

1. Chain Rule • If f and g are differentiable, then so is their composition f(g(x)) • Resulting differentiation formula called chain rule

2. First Approach

3. Examples • Compute the derivatives of the following functions:

4. Remark • Strategy always the same: find inside functionu and outside functionf to write complicated function as f(u). • Then use (chain rule) formula

5. More direct approach to chain rule • Derivative of f(u) is derivative of outside function evaluated at inside one times derivative of outside function:

6. Functions with parameters

7. Differentiating non-specified function

8. Examples involving multiple compositions • Apply chain rule repeatedly, one layer at a time until get to inner-most layer, e.g.

9. Implicit functions and implicit differentiation • y = f(x) defines y as a function of xexplicitly • Can also have y defined as a function of ximplicitly (in an equation) • E.g., y5x3 – 2yx + y2 = y3/4 • Can still differentiate using chain rule • Strategy: • Differentiate w.r.t. x both sides of equation, keeping in mind that y is a function of x • Solve the resulting equation for dy/dx

10. Examples

11. Related rates problems • Problems involving at least two changing quantities. Need to figure out the rate at which one is changing given sufficient information on all of the others. • Model problem: As two vehicles drive in different directions we should be able to deduce the speed at which they are separating if we know the individual speeds and directions.

12. Strategy for related rates • Draw a picture or otherwise make a mathematical model of the situation. • Label all quantities which can change as variables. • Identify the underlying independent variable in the problem. This is usually time but it need not be. • Identify in terms of the variables and their derivatives what is being asked and what is given. • Use the mathematical model to write down a relation among the variables. • Differentiate this relation with respect to the underlying independent variable, usually making heavy use of the chain rule. • Solve for the quantity wanted.

13. Model Problem • One vehicle starts driving north and one vehicle starts driving west from an intersection (as indicated by the arrows below). At the time the first vehicle is .3 miles north the intersection it is traveling at 20 mph. Simultaneously the second vehicle is .4 miles west of the intersection and is traveling at 25 mph. At that instant, how fast are the two vehicles separating?

14. Solution • Mathematical model: right triangle whose legs are increasing with time. Call legs x and y and hypotenuse z. All increasing with t = time (independent variable). • These three quantities are functions of t: x(t), y(t), z(t). • Know x, y, dx/dt, and dy/dt at particular time t0. Asked to find dz/dt • Differentiate Pythagorean equation z2 = x2 + y2: 2 x dx/dt + 2 y dy/dt = 2 z dz/dt • Given four of the six quantities (x, y, dx/dt, and dy/dt). Want dz/dt. Missing is only the quantity z, which we can compute using (.3)2 + (.4)2 = z2. • Get z = .5 • Therefore: 2 (.3) (20) + 2 (.4) (25) = 2 (.5) dz/dt • Gives answer dz/dt = 32 mph. • Answer:At the instant in question, the distance between the vehicles is increasing at 32 miles per hour.

15. Relative Growth rate of skulls vs. backbones • [skull length] = 1.162 [backbone length]0.933 • Question: How is growth rate of the backbone related to that of skull?

16. Another related rate problem • A stone is dropped into a pond, the ripples forming concentric circles which expand. • At what rate is the area of one of these circles increasing when the radius is 4 m and increasing at the rate of 0.5 m/s?

17. Higher derivatives • First derivative f  • Second derivative f  = (f )  • Higher order derivative notation: f (n)for nth order derivative. Also, dnf / dxn • Example: find second order derivative of x1/2 • Example: find d2y / dx2when x2 + y2 = 1