When gear A makes x turns, gear B makes u turns and gear C makes y turns., 3.6 Chain rule y turns ½ as fast as u u turns 3 times as fast as x So y turns 3/2 as fast as x Rates are multiplied

The Chain Rule for composite functions If y = f(u) and u = g(x) then y = f(g(x)) and multiply rates

Find the derivative (solutions to follow)

Solutions

Outside/Inside method of chain rule inside outside derivative of outside wrt inside derivative of inside think of g(x) = u

Outside/Inside method of chain rule example inside outside derivative of outside wrt inside derivative of inside

Outside/Inside method of chain rule inside outside derivative of outside wrt inside derivative of inside

Outside/Inside method of chain rule inside outside derivative of outside wrt inside derivative of inside

More derivatives with the chain rule product Simplify terms Combine with common denominator

More derivatives with the chain rule Quotient rule

The formulas for derivatives assume x is in radian measure. sin (x ° ) oscillates only  /180 times as often as sin (x) oscillates. Its maximum slope is  /180. Radians Versus Degrees d/dx[sin (x)] = cos (x) d/dx [ sin (x ° ) ] =  /180 cos (x ° )

3.7 Implicit Differentiation Although we can not solve explicitly for y, we can assume that y is some function of x and use implicit differentiation to find the slope of the curve at a given point y=f (x)

If y is a function of x then its derivative is y 2 is a function of y, which in turn is a function of x. using the chain rule: Find the following derivatives wrt x Use product rule

Implicit Differentiation 4. Solve for dy/dx 1.Differentiate both sides of the equation with respect to x, treating y as a function of x. This requires the chain rule. 2. Collect terms with dy/dx on one side of the equation. 3. Factor dy/dx

Find equations for the tangent and normal to the curve at (2, 4). Use Implicit Differentiation find the slope of the tangent at (2,4) find the slope of the normal at (2,4)

1.Differentiate both sides of the equation with respect to x, treating y as a function of x. This requires the chain rule. 2.Collect terms with dy/dx on one side of the equation. 3.Factor dy/dx 4. Solve for dy/dx Solution

Find dy/dx 1. Write the equation of the tangent line at (0,1) 2. Write the equation of the normal line at (0,1)

1.Differentiate both sides of the equation with respect to x, treating y as a function of x. This requires the chain rule. 2.Collect terms with dy/dx on one side of the equation. 3.Factor dy/dx 4. Solve for dy/dx Solution

Find dy/dx 1. Write the equation of the tangent line at (0,1) 2. Write the equation of the normal line at (0,1)

3.8 Higher Derivatives The derivative of a function f(x) is a function itself f ´(x). It has a derivative, called the second derivative f ´´(x) If the function f(t) is a position function, the first derivative f ´(t) is a velocity function and the second derivative f ´´(t) is acceleration. The second derivative has a derivative (the third derivative) and the third derivative has a derivative etc.

Find the second derivative for Find the third derivative for

In algebra we study relationships among variables The volume of a sphere is related to its radius The sides of a right triangle are related by Pythagorean Theorem The angles in a right triangle are related to the sides. In calculus we study relationships between the rates of change of variables. How is the rate of change of the radius of a sphere related to the rate of change of the volume of that sphere?

Examples of rates-assume all variables are implicit functions of t = time Rate of change in radius of a sphere Rate of change in volume of a sphere Rate of change in length labeled x Rate of change in area of a triangle Rate of change in angle, 3.9

Solving Related Rates equations 1.Read the problem at least three times. 2.Identify all the given quantities and the quantities to be found (these are usually rates.) 3.Draw a sketch and label, using unknowns when necessary. 4.Write an equation (formula) that relates the variables. 5.***Assume all variables are functions of time and differentiate wrt time using the chain rule. The result is called the related rates equation. 6.Substitute the known values into the related rates equation and solve for the unknown rate.

Figure 2.43: The balloon in Example 3. Related Rates A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the liftoff point. The angle of elevation is increasing at the rate of 0.14 rad/min. How fast is the balloon rising when the angle of elevation is is  /4? Given: Find:

Figure 2.43: The balloon in Example 3. Related Rates A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the liftoff point. At the moment the range finder’s elevation angle is  /4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?

Figure 2.44: Figure for Example 4. Related Rates A police cruiser, approaching a right angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. The cruiser is moving at 60 mph and the police determine with radar that the distance between them is increasing at 20 mph. When the cruiser is.6 mi. north of the intersection and the car is.8 mi to the east, what is the speed of the car? Given: Find:

Figure 2.44: Figure for Example 4. Given: Find: then s = 1

Figure 2.45: The conical tank in Example 5. Related Rates Water runs into a conical tank at the rate of 9 ft 3 /min. The tank stands point down and has a height of 10 ft and a base of radius 5 ft. How fast is the water level rising when the water is 6 ft. deep? Given: Find:

Figure 2.45: The conical tank in Example 5. Water runs into a conical tank at the rate of 9 ft 3 /min. The tank stands point down and has a height of 10 ft and a base of radius 5 ft. How fast is the water level rising when the water is 6 ft. deep? Given: Find: x=3

3..10 The more we magnify the graph of a function near a point where the function is differentiable, the flatter the graph becomes and the more it resembles its tangent. Differentiability

Differentiability and Linearization

Approximating the change in the function f by the change in the tangent line of f. Linearization

Write the equation of the straight line approximation Point-slope formula y=f(x)