3.6 Chain rule

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3.6 Chain rule
When gear A makes x turns, gear B makes u turns
and gear C makes y turns.,
u turns 3 times as fast as x
So y turns 3/2 as fast as x
y turns ½ as fast as u
Rates are multiplied
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The Chain Rule for composite functions
If y f(u) and u g(x) then y f(g(x)) and
multiply rates
multiply rates
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Find the derivative (solutions to follow)
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Solutions
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Solutions
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Outside/Inside method of chain rule
outside
inside
derivative of inside
derivative of outside wrt inside
think of g(x) u
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Outside/Inside method of chain rule example
outside
derivative of inside
inside
derivative of outside wrt inside
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Outside/Inside method of chain rule
outside
inside
derivative of inside
derivative of outside wrt inside
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Outside/Inside method of chain rule
outside
inside
derivative of inside
derivative of outside wrt inside
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More derivatives with the chain rule
product
Simplify terms
Combine with common denominator
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More derivatives with the chain rule
Quotient rule
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Radians Versus Degrees
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.
d/dxsin (x) cos (x)
d/dx sin (x) ?/180 cos (x)
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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
yf (x)
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y2 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
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Implicit Differentiation

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

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

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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)
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Solution

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

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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)
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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.
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Find the second derivative for
Find the third derivative for
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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?
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3.9
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,
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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.

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Related Rates
Figure 2.43 The balloon in Example 3.
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
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Related Rates
Figure 2.43 The balloon in Example 3.
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
finders 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?
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Related Rates
Figure 2.44 Figure for Example 4.
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
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Figure 2.44 Figure for Example 4.
Given
Find
then s 1
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Related Rates
Figure 2.45 The conical tank in Example 5.
Water runs into a conical tank at the rate of 9
ft3/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
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Figure 2.45 The conical tank in Example 5.
Water runs into a conical tank at the rate of 9
ft3/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
x3
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Differentiability
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.
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Differentiability and Linearization
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Linearization
Approximating the change in the function f by
the change in the tangent line of f.
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Write the equation of the straight line
approximation
Point-slope formula
yf(x)