2.1 tangents and derivatives at a point
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2.1 Tangents and Derivatives at a Point
Finding a Tangent to the Graph of a Function
To find a tangent to an arbitrary curve y=f(x) at a point P(x0,f(x0)), we• Calculate the slope of the secant through P and a nearby point Q(x0+h, f(x0+h)).• Then investigate the limit of the slope as h0.
Slope of the Curve
If the previous limit exists, we have the following definitions.
Reminder: the equation of the tangent line to the curve at P is Y=f(x0)+m(x-x0) (point-slope equation)
Example
(a) Find the slope of the curve y=x2 at the point (2, 4)?(b) Then find an equation for the line tangent to the curve there.
Solution
Rates of Change: Derivative at a Point
The expression
is called the difference quotient of f at x0 with increment h.
If the difference quotient has a limit as h approaches zero, that limit is named below.
( ) ( )f x h f xh
Summary
2.2 The Derivative as a Function
We now investigate the derivative as a function derived from f byConsidering the limit at each point x in the domain of f.
If f’ exists at a particular x, we say that f is differentiable (has a derivative) at x. If f’ exists at every point in the domain of f, we call fis differentiable.
Alternative Formula for the Derivative
An equivalent definition of the derivative is as follows. (let z = x+h)
Calculating Derivatives from the Definition
The process of calculating a derivative is called differentiation. It can be denoted by
Example. Differentiate
Example. Differentiate for x>0.
'( ) ( )df x or f xdx
( )f x x
2( )f x x
Notations
'( ) ' ( ) ( )( ) [ ( )]xdy df df x y f x D f x D f xdx dx dx
'( ) | | ( ) |x a x a x ady df df a f xdx dx dx
There are many ways to denote the derivative of a function y = f(x). Some common alternative notations for the derivative are
To indicate the value of a derivative at a specified number x=a, we use the notation
Graphing the Derivative
Given a graph y=f(x), we can plot the derivative of y=f(x) by estimating the slopes on the graph of f. That is, we plot the points (x, f’(x)) in the xy-plane and connect them with a smooth curve, which represents y=f’(x).
Example: Graph the derivative of the function y=f(x) in the figure below.
What we can learn from the graph of y=f’(x)?
If a function f is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval.
It is differentiable on a closed interval [a, b] if it is differentiable on the interior (a, b) and if the limits
exist at the endpoints.
0
( ) ( )limh
f a h f ah
0
( ) ( )limh
f b h f bh
Differentiable on an Interval; One-Sided Derivatives
Right-hand derivative at a
Left-hand derivative at b
A function has a derivative at a point if and only if the left-hand and right-hand derivatives there, and these one-sided derivatives are equal.
When Does A Function Not Have a Derivative at a Point
A function can fail to have a derivative at a point for several reasons, such as at points where the graph has
1. a corner, where the one-sided derivatives differ.
2. a cusp, where the slope of PQ approaches from one side and - from the other.
3. a vertical tangent, where the slope of PQ approaches from both sides or approaches - from both sides.
4. a discontinuity.
Differentiable Functions Are Continuous
Note: The converse of Theorem 1 is false. A function need not have a derivative at a point where it is continuous.
For example, y=|x| is continuous at everywhere but is not differentiable at x=0.
2.3 Differentiation Rules
The Power Rule is actually valid for all real numbers n.
Examples
Example.
Constant Multiple Rule
11 11 10 10
2 3 32
[ ] [ ] (11 ) 11
[ ] [ ] ( 2 ) 2
d dx x x xdx dxd d x x xdx x dx
Example.
Note: ( ) ( 1 ) 1 ( )d d d duu u udx dx dx dx
Derivative Sum Rule
Example.
Derivative Product Rule
[ ( ) ( )] ( ) '( ) ( ) '( )d f x g x f x g x g x f xdx
In function notation:
Example: 3 2Find if (2 2)(6 3 ).dy y x x xdx
Solution:
Example
Derivative Quotient Rule
2
( ) ( ) '( ) ( ) '( )[ ]( ) ( )
d f x g x f x f x g xdx g x g x
In function notation:
Example: 3 22 4Find '( ) if .
5x xy x yx
Solution:
Example
The derivative f’ of a function f is itself a function and hence may have a derivative of its own.
If f’ is differentiable, then its derivative is denoted by f’’. So f’’=(f’)’ and is called the second derivative of f.
Similarly, we have third, fourth, fifth, and even higher derivatives of f.
Second- and Higher-Order Derivatives
22 2
2
'''( ) ( ) '' ( )( ) [ ( )]xd y d dy dyf x y D f x D f xdx dx dx dx
A general nth order derivative can be denoted by
( ) ( 1)n
n n nn
d d yy y D ydx dx
Example: 3 2 y 4 2 6, thenIf x x x
2.4 The Derivative as a Rate of Change
Thus, instantaneous rates are limits of average rates.
When we say rate of change, we mean instantaneous rate of change.
Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk
Suppose that an object is moving along a s-axis so that we know its position s on that line as a function of time t: s=f(t). The displacement of the object over the time interval from t to t+∆t is ∆s = f(t+ ∆t)-f(t);
The average velocity of the object over that time interval is( ) ( )
avs f t t f tvt t
Velocity
To find the body’s velocity at the exact instant t, we take the limit of the Average velocity over the interval from t to t+ ∆t as ∆t shrinks to zero. The limit is the derivative of f with respect to t.
Besides telling how fast an object is moving, its velocity tells the direction ofMotion.
The speedometer always shows speed, which is the absolute value of velocity.Speed measures the rate of progress regardless of direction
The figure blow shows the velocity v=f’(t) of a particle moving on a coordinate line., what can you say about the movement ?
Acceleration
The rate at which a body’s velocity changes is the body’s acceleration. The acceleration measures how quickly the body picks up or loses speed.A sudden change in acceleration is called a jerk.
Example
Near the surface of the earth all bodies fall with the same constant acceleration. In fact, we have
s=(1/2)gt2 ,
where s is the distance fallen and g is the acceleration due to Earth’s gravity.
With t in seconds, the value of g at sea lever is 32 ft/ sec2 or 9.8m/sec2.
Example
Example: Figure left shows the free fall of a heavy ball bearing released from rest at time t=0.(a)How many meters does the ball fall in the first 2 sec?
(b)What is its velocity, speed, and acceleration when t=2?
2.5 Derivatives of Trigonometric Functions
Example: Find if cos .dy y x xdx
Solution:
Example
Example: cosFind if .
1 sindy xydx x
Solution:
Example
Example: A body hanging from a spring is stretched down 5 units beyondIts rest position and released at time t=0 to bob up and down. Its position at any later time is s=5cos t. What are its velocity and acceleration at time t?
sin cos 1 1tan , cot , sec , csccos sin cos sinx xx x x xx x x x
Since
We have
Example: Find '' if ( ) tan .y f x x
Solution:
Example
2.6 Exponential Functions
In general, if a1 is a positive constant, the function f(x)=ax is the exponential function with base a.
If x=n is a positive integer, then an=a a … a.
If x=0, then a0=1,
If x=-n for some positive integer n, then
If x=1/n for some positive integer n, then
If x=p/q is any rational number, then
If x is an irrational number, then
1 1( )n nnaa a
1/n na a
/ ( )q qp q p pa a a
Rules for Exponents
The Natural Exponential Function ex
The most important exponential function used for modeling natural, physical, and economic phenomena is the natural exponential function, whose base is a special number e.
The number e is irrational, and its value is 2.718281828 to nine decimal places.
The graph of y=ex has slope 1 when it crosses the y-axis.
Derivative of the Natural Exponential Function
Example. Find the derivative of y=e-x.
Solution:
Example. Find the derivative of y=e-1/x.
2.7 The Chain Rule
Example: 2Let y= sin( ). Find .
dxdx
Solution:
Example
“Outside-inside” Rule
It sometimes helps to think about the Chain Rule using functional notation. If y=f(g(x)), then
In words, differentiate the “outside” function f and evaluate it at the “inside” function g(x) left alone; then multiply by the derivative of the “inside” function.
'( ( )) '( )dy f g x g xdx
Example
Example. Differentiate sin(2x+ex) with respect to x.
Solution.
Example. Differentiate e3x with respect to x.
Solution.
In general, we have
For example.
Example: find derivative of |x| when x ≠ 0.
sin sin sin( ) (sin ) cosx x xd de e x e xdx dx
Repeated Use of the Chain Rule
Sometimes, we have to apply the chain rule more than once to calculate a derivative.
Find [sin(tan3 )].d xdx
Example.
Solution.
The Chain Rule with Powers of a Function
If f is a differentiable function of u and if u is a differentiable function of x, then substituting y = f(u) into the Chain Rule formula leads to the formula
( ) '( )d duf u f udx dx
This result is called the generalized derivative formula for f.
For example. If f(u)=un and if u is a differentiable function of x, then we canObtain the Power Chain Rule:
1n nd duu nudx dx
Example: 8Find ( 2)d xdx
Solution:
Example
Example: Find [ tan ].d xdx
Solution:
Example
Example: 3 10Find [(1 sec ) ]d x
dx
Solution:
Example
2.8 Implicit Differentiation
Definition. We will say that a given equation in x and y defines the function f implicitly if the graph of y = f(x) coincides with a portion of the graph of the equation.
Example:
•The equation implicitly defines functions
2 2 1x y 2 2
1 2( ) 1 and ( ) 1f x x f x x
2x y
1 2( ) and ( )f x x f x x
•The equation implicitly defines the functions
There are two methods to differentiate the functions defined implicitly by the equation.
For example: Find / if 1dy dx xy
One way is to rewrite this equation as , from which it
follows that
1yx
2
1dydx x
Two differentiable methods
With this approach we obtain [ ] [1]
[ ] [ ] 0
0
d dxydx dxd dx y y xdx dxdyx ydxdy ydx x
The other method is to differentiate both sides of the equation before solving for y in terms of x, treating y as a differentiable function of x. The method is called implicit differentiation.
Since , 1yx
2
1dydx x
Two differentiable methods
Implicit Differentiation
Example: Use implicit differentiation to find dy / dx if
Solution:
2 2 3x y x
Example
Example: Find dy / dx if 3 3 11 0y x
Solution:
Example
Lenses, tangents and Normal Lines
In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light makes with the line perpendicular to the surface of the lens at the point of entry.
This line is called the normal to the surf surface at the point of entry.
The normal is the line perpendicular to the tangent of the profile curve at the point of entry.
Example
Show that the point (2, 4) lies on the curve x3+y3-9xy=0. Then find the tangent and normal to the curve there.
Derivatives of Higher Order
Find dy2 /dx2 if 2x3-3y2=8.
2.9 Inverse Functions and Their Derivatives
A function that undoes, or inverts, the effect of a function f is called the inverse of f.
Examples
Inverse Function
Note the symbol f -1 for the inverse of f is read “f inverse”. The “-1” in f -1 isnot an exponent; f -1 (x) does not mean 1/f(x).
Finding Inverses
The process of passing from f to f -1 can be summarized as a two-step process.
1. Solve the equation y=f(x) for x. This gives f formula x=f -1(y) where x is expressed as a function of y.
2. Interchange x and y, obtaining a formula y=f -1(x), where f -1 is expressed in the conventional format with x as the independent
variable and y as the dependent variable.
Examples
Find the inverse of (a) y=3x-2.(b) y=x2,x≥0.Solution:
Derivative Rule for Inverses
Derivative Rule for Inverses
Example
Let f(x)= x3-2. Find the value of df-1/dx at x=6 = f(2) without finding a formula for f -1 (x).
2.10 Logarithmic Functions
Natural Logarithm Function
Logarithms with base e and base 10 are so important in applications thatCalculators have special keys for them. logex is written as lnx log10x is written as logx
The function y=lnx is called the natural logarithm function, and y=logx isOften called the common logarithm function.
Properties of Logarithms
Properties of ax and logax
Derivative of the Natural Logarithm Function
3Find [ln( 4)]d xdx
Example:
Solution:
Note: 1[ln ] , 0d x xdx x
Since y=lnx is the inverse function of y=ex, we have
Example
Example: Find [ln | cos |]d xdx
Solution:
Furthermore, since |x|=x when x>0 and |x|= -x when x<0,
Derivatives of au
Example:
Note that [ ] ln , [ ] , [ ]x x x x u ud d d dua a a e e e edx dx dx dx
Since ax=exlna, we can find the following result.
Derivatives of logau
Example:
Note that 1lnd duudx u dx
Since logax =lnx/lna, we can find the following result.
Logarithmic Differentiation
The derivatives of positive functions given by formulas that involve products, quotients, and powers can often be found more quickly if we take the natural logarithm of both sides before differentiating. This process is called logarithmic differentiation.
Example. Find dy/dx if 2 1/2( 1)( 3) , 1
1x xy x
x
The Number e as a Limit
2.11 Inverse Trigonometric Functions
The six basic trigonometric functions are not one-to-one (their values Repeat periodically). However, we can restrict their domains to intervals on which they are one-to-one.
Six Inverse Trigonometric Functions
Since the restricted functions are now one-to-one, they have inverse, which we denoted by
These equations are read “y equals the arcsine of x” or y equals arcsin x” and so on.
Caution: The -1 in the expressions for the inverse means “inverse.” It doesNot mean reciprocal. The reciprocal of sinx is (sinx)-1=1/sinx=cscx.
1
1
1
1
1
1
sin arcsin
cos arccos
tan arctan
cot arccot
sec arcsec
csc arccsc
y x or y x
y x or y x
y x or y x
y x or y x
y x or y x
y x or y x
Derivative of y = sin-1x
Example: Find dy/dx if
Solution:
1 2sin ( )y x
Derivative of y = tan-1x
Example: Find dy/dx if
Solution:
1tan ( )xy e
Derivative of y = sec-1x
Example: Find dy/dx if
Solution:
1 3sec (4 )y x
Derivative of the other Three
There is a much easier way to find the other three inverse trigonometric Functions-arccosine, arccotantent, and arccosecant, due to the followingIdentities:
It follows easily that the derivatives of the inverse cofunctions are the negativesof the derivatives of the corresponding inverse functions.
2.13 Linearization and Differentials
In general, the tangent to y=f(x) at a point x=a, where f is differentiable, passes through the point (a, f(a)), so its point-slope equation is y=f(a)+f’(a)(x-a).
Thus this tangent line is the graph of the linear function L(x)=f(a)+f’(a)(x-a)..For as long as this line remains close to the graph of f, L(x) gives a goodapproximation to f(x).
Linearization
Example
Find the linearization of f(x)=cosx at x=π/2.
Also an important linear approximation for roots and poewrs is
(1+x)k==1+kx (x near 0; any number k).
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