CHAPTER 2

THE DERIVATIVE

2.1

• Two Problems with One Theme

Tangent Lines & Secant Lines• The slope of a secant line between 2 points

on a curve is the change in y-values divided by the change in x-values.

• Since a tangent line touches only one point on the curve, how do we find the slope of the line? We consider the slope of 2 points that are INFINITELY close together at the point of tangency…thus a limit!

Average Velocity & Instantaneous Velocity

• Similar to slope of a secant line, to find average velocity, we find the change in distance divided by the change in time between 2 points on a time interval.

• To find instantaneous velocity, we find the difference in distance and time between two points in time that are INIFINITELY close together…again, a limit!

Tangent Line Slope at x = c & Instantaneous Velocity at t = c

are defined the SAME

h

cfhcfvm

h

)()(lim

0tan

A falling body’s velocity is defined. Find the instantaneous velocity at t = 3

seconds.

sec)/963(32)1632(

lim

1632lim

16163216lim

16)2(16lim

16)(16lim

16

0

2

0

222

0

222

0

22

0

2

ftvtth

hthh

hth

h

ththt

h

ththt

h

tht

tv

h

hh

hh

2.2 The Derivative

• The derivative of f(x) is designated as f’(x) or f’ or y’.

h

xfhxfxfy

h

)()(lim)(''

0

Differentiability implies continuity.

• If the graph of a function has a tangent at point c, then there is no “jump” on the graph at that point, thus is continuous there.

Find f’(x).

)cos()sin(

lim)cos()1)(cos(

lim)sin(

)sin()cos()1))(cos(sin(lim

1sin1)sin()cos()cos()sin(lim

]1[sin]1)[sin(lim)('

1sin)(

00

0

0

0

xh

hx

h

hx

h

hxhxh

xhxhxh

xhxxf

xxf

hh

h

h

h

2.3

• Rules for Finding Derivatives

A derivative is a limit.

• Therefore, the rules for limits, essentially become the rules for derivatives.

• Derivative of a sum/difference is the sum/difference of the derivatives.

• Derivative of a product/quotient is the product/quotient of the derivatives.

Product Rule

• If f and g are differentiable functions, then

)(')()(')()()'( xfxgxgxfxgf

Quotient Rule

• Let f & g be differentiable functions with g(x) not equal 0.

2))((

)(')()(')()('

xg

xgxfxfxgx

g

f

Differentiate the following

2

2

2

23

2

3223

2

32

2

322

2

22

2

)2(

)2586(2

)2(

501612'

)2(

10640362018'

)2(

)106()2018)(2('

)2(

)106()]6()2012)[(2('

)2(

)1)(2)(53()]3)(2()4)(53)[(2('

2

)2)(53(

x

xxx

x

xxxy

x

xxxxxxy

x

xxxxxy

x

xxxxxxy

x

xxxxxxy

x

xxy

2.4

• Derivatives of Trigonometric Functions

f’(sin x ) = cos xf’(cos x) = - sin x

• Find derivatives of other trig. functions using these derivatives and applying product rule and/or quotient rule

xxx

xxxf

x

xxxxxf

x

xx

222

22

2

sec)(cos

1

)(cos

)(sin)(cos)(tan'

)(cos

))sin()(sin()cos()cos()(tan'

)cos(

)sin()tan(

Derivatives of sec(x), csc(x) and cot(x)

• All are found by applying the product and/or quotient rules and using known derivatives of sin(x) and cos(x).

xxf

xxxf

xxxf

2csc)(cot'

cotcsc)(csc'

tansec)(sec'

Find the derivative of the following

2

2

2

2

cotseccsctansec'

)1)(cot(sec)csctan(sec'

cotsec

x

xxxxxxxy

x

xxxxxxy

x

xxy

2.5

• The Chain Rule

For a composite function, its derivative is found by taking the

derivative of the outer function, with respect to the inner function, times the derivative of the inner function

with respect to x.

• If the composition consists of 3 or more functions, continue to take the derivative of the next inner function, with respect to the function within it, until, finally, the derivative is taken with respect to x.

Find the derivative (note this is the composition of 3 functions,

therefore there will be 3 “pieces” to the chain.)

)cotcsc3(])csc(5[])csccos[('

])cscsin[(24353

53

xxxxxxxy

xxy

2.6

• Higher-Order Derivatives

f’’=2nd derivativef’’’=3rd derivative

f’’’’=4th derivative, etc…

• The 2nd derivative is the derivative of the 1st derivative.

• The 3rd derivative is the derivative of the 2nd derivative, etc.

Velocity is the derivative of distance with respect to time (1st derivative)

and Acceleration is the derivative of velocity with respect to time (2nd

derivative of distance with respect to time)

• Up (or right) is a positive velocity.

• Down (or left) is a negative velocity.

• When an object reaches its peak, its velocity equals zero.

2.7

• Implicit Differentiation

• (An application of the chain rule!)

• y is now considered as a function of x, therefore we apply the chain rule to y

• Apply all appropriate rules and solve for dy/dx.

Find the derivative

yxyx

xyy

dx

dy

xyyyxyxdx

dydx

dyyxyy

dx

dyxxy

dx

dyyy

dx

dyxxy

xyxy

2

2

2

2

sec)cos(

)cos(2

)cos(2)sec)cos((

2)(sec)cos()cos(

2)(sec)1()cos(

2)tan()sin(

2.8

• Related Rates

• A very, very important application of the derivative!

• Applies to situations where more than one variable is changing with respect to time.

• The other variables are defined with respect to time, and we differentiate implicitly with respect to time.

A rock is dropped in a pond and the area of the circle and the radius of the circle formed are both changing with

respect to time. The radius is changing at a constant rate of 2 cm/sec. How

fast is the area changing 4 sec after the rock is dropped?(radius would be 8 cm)

sec201

sec64

sec88

sec8

sec422

2

2

2

cm

dt

dA

cmcmcm

cmr

cmr

dt

drr

dt

dA

rA

2.9

• Differentials & Approximations

• dx is the differential of x, graphically it is the change in the x of the tangent to the curve (dy/dx)

• dy is the differential of y, graphically is corresponds to the change in the y of the tangent to the curve (dy/dx)

Differentials can be used to approximate function values, for which you know the

value evaluated at a point nearby.

2.510

252)25(')25()225(

2

1)('

2,25,)(,27

)(')()(

fff

xxf

xxxxfFind

xxfxfxxf