1. definition of derivative 2. derivatives as functions 3...

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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 .