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Lecture4 | 1 Chapter 2 Derivatives Outline 1. Definition of derivative 2. Derivatives as functions 3. Differentiation rules 4. Chain rule 5. Higher derivatives
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Chapter 2 Derivatives Outline 1. Definition of derivative 2. Derivatives as functions 3. Differentiation rules 4. Chain rule 5. Higher derivatives
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We will study a special type of limit:
which is called the derivative of with respect to at .
The operation to get this limit is called differentiation.
If the above limit exists, it defines a new function called the derivative of .
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a. Tangent Lines
The slope to at is
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EX Find the slope of the tangent line to the curve at the point .
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b. Velocity
The (instantaneous) velocity at time of an object with displacement function at time is
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EX An object is moving along a line with the displacement function . Find the velocity at the instance .
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1.1. Definition of Derivative Def For a function and a number , the derivative of at is the number
provided the limit exists.
It is also denoted by
If exists, is called differentiable at , otherwise, not differentiable at .
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EX Find if
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EX Let
Determine whether is continuous or differentiable at ?
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EX Determine whether is differentiable at , if .
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Derivatives as Rates of Change
Suppose is a quantity that depends on another quantity written by .
Let and . Then
is called the change in , and
is called the change in .
The quotient
is called the average rate of change of .
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The limit
is called the rate of change of at . The above limit is precisely . So
is the rate of change of at . Remark Other notation
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EX In an electrical circuit, the amount of charges circulating at time is a function . The rate of change of is called the current and is denoted by . Thus
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1.2. Derivative as a Function
Def Let be a function. Define another function by
for every where the limit exists.
is called the derivative of .
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EX (Use definition) If , find the formula for the derivative .
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EX (Graph has corner) Is the function
differentiable at ?
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Theorem If is differentiable at , then is continuous at . [A function continuous at may not be differentiable at ! The the previous EX.]
If is not continuous at then it is not differentiable at .
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EX (Graph has a discontinuity) Explain why the function
fails to be differentiable at ?
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Differentiation Rules
Sum
Subtraction
Constant multiple
Constant
Powers (including root)
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EX (Use differentiation rules) Find the derivative of
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Product
Quotient
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EX Differentiate the following functions.
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EX If
, , calculate .
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EX Find the tangent line to the curve
at the point .
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EX The position function of a particle is given by
When does the particle reach a velocity of 5 m/s?
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Chain Rule
Chain Rule If then
Equivalently,
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EX If find .
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EX Find and
if
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EX Suppose ,
and . Find .
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EX Show that
and
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Higher Derivatives Def For a differentiable function , we can differentiate to get the first derivative:
If, furthermore, is also differentiable, we can differentiate one more time to get
This function is obtained from by
differentiating twice, it is called the second derivative of , denoted by
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EX Let . Find the first and second derivatives of .
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EX (Second derivative as acceleration) If an object is moving along a line with displacement function
1. Find the velocity and acceleration of the object after 4 s. 2. When is the object speeding up? When is it slowing down?
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Def From the second derivative , if it is differentiable then we can define
which is called the third derivative of . It is often denoted
Similarly, one can define the ’th
derivative of which is obtained by differentiating times to :
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EX Let . Find
for any positive integer .
