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Index FAQ
The derivative as the slope of the tangent line
(at a point)
Index FAQ
Video help: MIT!!! http://ocw.mit.edu/courses/
mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/
Index FAQ
What is a derivative?
A function, which gives the:the rate of change of a
function in generalthe slope of the line tangent
to the curve in general
Index FAQ
What is a differential quotient?
Just a number!the rate of change of a function at a
given pointthe slope of the line tangent to the
curve at a certain pointThe substitutional value of the
derivative
Index FAQ
The tangent line
single pointof intersection
Index FAQ
slope of a secant line
ax
f(x)
f(a)
f(a) - f(x)
a - x
Index FAQ
slope of a (closer) secant line
ax
f(x)
f(a)
f(a) - f(x)
a - x
x
Index FAQ
closer and closer…
a
Index FAQ
watch the slope...
Index FAQ
watch what x does...
ax
Index FAQ
The slope of the secant line gets closer and closer to the slope of the tangent line...
Index FAQ
As the values of x get closer and closer to a!
ax
Index FAQ
The slope of the secant lines gets closer
to the slope of the tangent line...
...as the values of x get closer to a
Translates to….
Index FAQ
limax
f(x) - f(a)x - a
Equation for the slope
Which gives us the the exact slope of the line tangent to the curve at a!
as x goes to a
Differential quotient
Index FAQ
Differential quotient: other form
aa+h
f(a+h)
f(a)
f(x+h) - f(x)
(x+h) - x= f(x+h) - f(x)
h
(For this particular curve, h is a negative value)
h
limh0
Index FAQ
Rates of Change:
Average rate of change = f x h f x
h
Instantaneous rate of change = 0
limh
f x h f xf x
h
These definitions are true for any function.
Velocity and other Rates of Change
Index FAQ
Consider a graph of displacement (distance traveled) vs. time.
time (hours)
distance(miles)
Average velocity can be found by taking:change in position
change in time
s
t
t
sA
B
ave
f t t f tsV
t t
The speedometer in your car does not measure average velocity, but instantaneous velocity.
0
limt
f t t f tdsV t
dt t
(The velocity at one moment in time.)
Velocity and other Rates of Change- physical menaing of the differential quotient
Index FAQ
Velocity and other rates of change
Velocity is the first derivative of position.
Acceleration is the second derivative of position.
Index FAQ
Example:Free Fall Equation
21
2s g t
GravitationalConstants:
2
ft32
secg
2
m9.8
secg
2
cm980
secg
2132
2s t
216 s t 32 ds
V tdt
Speed is the absolute value of velocity.
Velocity
Index FAQ
Acceleration is the derivative of velocity.
dva
dt
2
2
d s
dt example:
32v t
32a If distance is in: feet
Velocity would be in:feet
sec
Acceleration would be in:
ftsec sec
2
ft
sec
3.4 Velocity and other Rates of Change
Index FAQtime
distance
acc posvel pos &increasing
acc zerovel pos &constant
acc negvel pos &decreasing
velocityzero
acc negvel neg &decreasing acc zero
vel neg &constant
acc posvel neg &increasing
acc zero,velocity zero
Velocity and other Rates of Change
Index FAQ
To be differentiable, a function must be continuous and smooth.Derivatives will fail to exist at:
corner
f x x
cusp
2
3f x x
vertical tangent
3f x x
discontinuity
1, 0
1, 0
xf x
x
Differentiability
Index FAQ
Theorem : f is differentiable on the interval (a,b). f is continuous on the interval (a,b).
Proof: Assume that f ’(c) exists for any c in (a,b).
Then lim [ f(c+h)- f(c)] . h0
= f ’(c) • 0 = 0
f ’(c) = lim f(c+h) -f(c) . h 0 h
= f ’(c) • lim h . h 0
So lim [ f(c+h) - f(c)] = 0 . h0
, and from here we get lim f(c+h) = f(c) . . h0
So f is continuous at c for every c in (a,b).
/ • h
Index FAQ
Example: Since the derivative of f(x)= 5x2+x+1 is f ’(x) = 10x+1, which exists for every real number x. So f(x)= 5x2+x+1 is continuous everywhere.
RemarkThe reverse of this theorem is not
true.
Counter example: We know that f(x) = |x| is continuous on R , but at x=0 it’s not differentiable since:
lim l0+hl –l0l h 0 h
= lim lhl . h 0 h
, which approaches to +1 if h 0 –1 if h0
Index FAQ
To be differentiable, a function must be continuous and smooth.Derivatives will fail to exist at:
corner
f x x
cusp
2
3f x x
vertical tangent
3f x x
discontinuity
1, 0
1, 0
xf x
x
Differentiability
Index FAQ
If the derivative of a function is its slope, then for a constant function, the derivative must be zero. There is no change...
0dc
dx
example: 3y
0y
The derivative of a constant is zero.
Derivatives of some elementary functions
Index FAQ
We saw that if , .2y x 2y x
This is part of a pattern.
1n ndx nx
dx
examples:
4f x x
34f x x
8y x
78y x
power rule
Derivatives of some elementary functions
Index FAQ
Find the horizontal tangents of:
4 22 2y x x 34 4
dyx x
dx
Horizontal tangents occur when slope = zero.
34 4 0x x 3 0x x
2 1 0x x
1 1 0x x x
0, 1, 1x
Substituting the x values into the original equation, we get:
2, 1, 1y y y
(The function is even, so we only get two horizontal tangents.)
Rules for Differentiation
Index FAQ
Rates of Change:
Average rate of change = f x h f x
h
Instantaneous rate of change = 0
limh
f x h f xf x
h
These definitions are true for any function.
( x does not have to represent time. )
Velocity and other Rates of Change
Index FAQ
2
0
2
Consider the function siny
We could make a graph of the slope: slope
1
0
1
0
1Now we connect the dots!The resulting curve is a cosine curve.
sin cosd
x xdx
Derivatives of Trigonometric Functions
Index FAQ
Derivatives of Trigonometric Functions
h
xsin)hxsin(lim)'x(sin
0h
h
xsinxcoshsinhcosxsinlim
0h
h
xcoshsinlim
h
)1h(cosxsinlim
0h0h
h
xcoshsin)1h(cosxsinlim
0h
Proof
h
xh
h
hxx
dx
dhh
cossinlim
)1(cossinlimsin
00
Index FAQ
Derivative of the cosine Function
h
xcos)hxcos(lim)'x(cos
0h
h
xsinhsinlim
h
)1h(cosxcoslim
0h0h
h
xsinhsin)1h(cosxcoslim
0h
Find the derivative of cos x:
h
xsinhsinlim
h
)1h(cosxcoslim
0h0h
Index FAQ
Derivative of the cosine function is sine (cont.)
xsin1.xsin0.xcosh
hsinlimxsin
h
)1h(coslimosxc
h
xsinhsin
h
)1h(cosxcoslim
h
xsinhsin)1h(cosxcoslim
h
xcosxsinhsinhcosxcoslim
0h0h
0h
0h
0h
Index FAQ
We can find the derivative of tangent x by using the quotient rule.
tand
xdx
sin
cos
d x
dx x
2
cos cos sin sin
cos
x x x x
x
2 2
2
cos sin
cos
x x
x
2
1
cos x2sec x
2tan secd
x xdx
Derivatives of Trigonometric Functions
Index FAQ
Derivatives of the remaining trig functions can be determined the same way.
sin cosd
x xdx
cos sind
x xdx
2tan secd
x xdx
2cot cscd
x xdx
sec sec tand
x x xdx
csc csc cotd
x x xdx
Derivatives of Trigonometric Functions
Index FAQ
The Derivatives of the Sum, Difference, Product and Quotient
If and are derivable, and is any constant, u x v x C
then so is , , , and
. Its derivative is given by the formula
u x v x u x v x Cu x
u x
v x
(1) ( ) ( ) ( ) ( )u x v x u x v x
(2) ( ) ( ) ( ) ( ) ( ) ( )u x v x u x v x u x v x
(3) ( ) ( )Cu x Cu x 2
( ) ( ) ( ) ( ) ( )(4) ( ( ) 0)
( ) ( )
u x u x v x u x v xv x
v x v x
Index FAQ
Proof
(1) Let ( ) ( ), we have to examiney u x v x
0 0
( ) ( ) ( ) ( )lim limx x
y u x x v x x u x v x
x x
0
( ) ( ) ( ) ( )limx
u x x u x v x x v x
x
(1) ( ) ( ) ( ) ( )u x v x u x v x
Index FAQ
0lim( ) ( ) ( )x
u vu x v x
x x
Thus ( ) ( ) is derivable and
( ) ( ) ( ) ( )
u x v x
u x v x u x v x
A similar argument applies to ( ) ( ),
that is
( ) ( ) ( ) ( )
u x v x
u x v x u x v x
(1) ( ) ( ) ( ) ( )u x v x u x v x
Index FAQ
x 0
(2) Let ( ) ( ), then we express in terms
of and . Finally, we determine by
examining lim
y u x v x y
u v y x
y
x
0 0
( ) ( ) ( ) ( )lim limx x
y u x x v x x u x v xy
x x
0
[ ( ) ][ ( ) ] ( ) ( )limx
u x u v x v u x v x
x
(2) ( ) ( ) ( ) ( ) ( ) ( )u x v x u x v x u x v x
Index FAQ
0
( ) ( )limx
u x v v x u u v
x
0lim[ ( ) ( ) ]x
u v vv x u x ux x x
( ) ( ) ( ) ( )u x v x u x v x
Thus, ( ) ( ) is derivable and
( ) ( ) ( ) ( ) ( ) ( )
u x v x
u x v x u x v x u x v x
Index FAQ
HOMEWORK!!
(3) ( ) ( )Cu x Cu x
Index FAQ
2
( ) ( ) ( ) ( ) ( )(4) ( ( ) 0)
( ) ( )
u x u x v x u x v xv x
v x v x
0 0
( ) ( )( ) ( )
lim limx x
u x x u xy v x x v x
yx x
0
( ) ( )( ) ( )
limx
u x u u xv x v v x
x
Index FAQ
0 0
( ) ( )( ) ( )
lim limx x
u x x u xy v x x v x
yx x
0
( ) ( )( ) ( )
limx
u x u u xv x v v x
x
2
( ) ( ) ( ) ( ) ( )(4) ( ( ) 0)
( ) ( )
u x u x v x u x v xv x
v x v x
Index FAQ
0
[ ( ) ] ( ) ( )[ ( ) ]lim
[ ( ) ] ( )x
u x u v x u x v x v
v x v v x x
0
( ) ( )lim
[ ( ) ] ( )x
uv x u x v
v x v v x x
0
( ) ( )lim
[ ( ) ] ( )x
u vv x u xx xv x v v x
Index FAQ
dy dy du
dx du dx
Chain Rule:
example: sinf x x 2 4g x x Find: at 2f g x
cosf x x 2g x x 2 4 4 0g
0 2f g cos 0 2 2 1 4 4
Chain Rule
If is the composite of and , then:f g y f u u g x
at at xu g xf g f g )('))((' xgxgf
Index FAQ
Remark
f(g(x))’= f ’(g(x)) g’(x) says that to get the
derivative of the “nested functions” you multiply
the derivative of each one starting from left to
right and so on
Index FAQ
Example : Find y’(1) for y = (3x2-2)3( 5x3-x-3)4
y ’= 3(3x2 -2)2 (3x2-2)’( 5x3-x-3)4 + (3x2-2)3 4( 5x3-x-3)3 ( 5x3-x-3)’
y ’= 3(3x2 -2)2 (6x) ( 5x3-x-3)4 + (3x2-2)3 4( 5x3-x-3)3 (15x2-1)
y ’(1) = 3(3-2)2 (6) (5-1-3)4 + (3-2)3 4 (5-1-3)3 (15-1)
YOUR TURN!, find when x=1 .
= 74
dy .
dx
. 2 x - 1 √5x2+4
For y =
Example for using Chain rule
Index FAQ
2sin 4y x
2 2cos 4 4d
y x xdx
2cos 4 2y x x
Example for using Chain rule
Index FAQ
2cos 3d
xdx
2cos 3
dx
dx
2 cos 3 cos 3d
x xdx
2cos 3 sin 3 3d
x x xdx
2cos 3 sin 3 3x x
6cos 3 sin 3x x
The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)
Example for using Chain rule
Index FAQ
2 2 1x y This is not a function, but it would still be nice to be able to find the slope.
2 2 1d d dx y
dx dx dx Do the same thing to both sides.
2 2 0dy
x ydx
Note use of chain rule.
2 2dyy xdx
2
2
dy x
dx y
dy x
dx y
Implicit Differentiation
Index FAQ
22 siny x y 22 sin
d d dy x y
dx dx dx
This can’t be solved for y.
2 2 cosdy dy
x ydx dx
2 cos 2dy dy
y xdx dx
22 cosdy
xydx
2
2 cos
dy x
dx y
This technique is called implicit differentiation.
1 Differentiate both sides w.r.t. x.2 Solve for y’
Implicit Differentiation
Index FAQ
Implicit Differentiation
Implicit Differentiation Process
1. Differentiate both sides of the equation with respect to x.
2. Collect the terms with y’=dy/dx on one side of the equation.
3. Factor out y’=dy/dx .
4. Solve for y’=dy/dx .
Index FAQ
Find the equations of the lines tangent and normal to the
curve at .2 2 7x xy y ( 1, 2)
2 2 7x xy y
2 2 0dydy
x yx ydxdx
Note product rule.
2 2 0dy dy
x x y ydx dx
22dy
y xy xdx
2
2
dy y x
dx y x
2 2 1
2 2 1m
2 2
4 1
4
5
Implicit Differentiation
Index FAQ
Find the equations of the lines tangent and normal to the
curve at .2 2 7x xy y ( 1, 2)
4
5m tangent:
42 1
5y x
4 42
5 5y x
4 14
5 5y x
normal:
52 1
4y x
5 52
4 4y x
5 3
4 4y x
Implicit Differentiation
Index FAQ
Find if .2
2
d y
dx3 22 3 7x y
3 22 3 7x y 26 6 0x y y
26 6y y x 26
6
xy
y
2x
yy
2
2
2y x x yy
y
2
2
2x xy y
y y
2 2
2
2x xy
y
x
yy
4
3
2x xy
y y
Substitute back into the equation.
y
Implicit Differentiation
Index FAQ
siny x
1siny xWe can use implicit differentiation to find:
1sind
xdx
1siny x
sin y xsin
d dy x
dx dx
cos 1dyydx
1
cos
dy
dx y
Derivatives of Inverse Trigonometric Functions
Index FAQ
We can use implicit differentiation to find:
1sind
xdx
1siny x
sin y xsin
d dy x
dx dx
cos 1dyydx
1
cos
dy
dx y
2 2sin cos 1y y
2 2cos 1 siny y 2cos 1 siny y
But2 2
y
so is positive.cos y
2cos 1 siny y
2
1
1 sin
dy
dx y
2
1
1
dy
dx x
Derivatives of Inverse Trigonometric Functions
Index FAQ
1siny x
1
cos
dy
dx y
Derivatives of Inverse Trigonometric Functions
)cos(sin
11 xdx
dy
21
1
xdx
dy
xy sin
1cos dx
dyy
Index FAQ
Derivatives of Inverse Trigonometric Functions
)(tansec
112 xdx
dy
21
1
xdx
dy
xy tan
1sec2 dx
dyy
Find xdx
d 1tan
xy 1tan
ydx
dy2sec
1
Index FAQ
Look at the graph of xy e
The slope at x = 0 appears to be 1.
If we assume this to be true, then:
0 0
0lim 1
h
h
e e
h
definition of derivative
Derivatives of Exponential and Logarithmic Functions
Index FAQ
Now we attempt to find a general formula for the derivative of using the definition.
xy e
0
limx h x
x
h
d e ee
dx h
0lim
x h x
h
e e e
h
0
1lim
hx
h
ee
h
0
1lim
hx
h
ee
h
1xe xe
This is the slope at x = 0, which we have assumed to be 1.
Derivatives of Exponential and Logarithmic Functions
Index FAQ
xe is its own derivative!
If we incorporate the chain rule: u ud due e
dx dx
We can now use this formula to find the derivative ofxa
Derivatives of Exponential and Logarithmic Functions
Index FAQ
xda
dx
ln xade
dx
lnx ade
dx ln lnx a d
e x adx
Incorporating the chain rule:
lnu ud dua a a
dx dx
Derivatives of Exponential and Logarithmic Functions
aaadx
d xx ln
Index FAQ
So far today we have:
u ud due e
dx dx lnu ud du
a a adx dx
Now it is relatively easy to find the derivative of .ln x
Derivatives of Exponential and Logarithmic Functions
Index FAQ
lny xye x
yd de x
dx dx
1y dyedx
1y
dy
dx e
1ln
dx
dx x
1ln
d duu
dx u dx
Derivatives of Exponential and Logarithmic Functions
Index FAQ
To find the derivative of a common log function, you could just use the change of base rule for logs:
logd
xdx
ln
ln10
d x
dx
1ln
ln10
dx
dx
1 1
ln10 x
The formula for the derivative of a log of any base other than e is:
1log
lna
d duu
dx u a dx
Derivatives of Exponential and Logarithmic Functions
Index FAQ
u ud due e
dx dx lnu ud du
a a adx dx
1log
lna
d duu
dx u a dx
1ln
d duu
dx u dx
Derivatives of Exponential and Logarithmic Functions
Index FAQ
Derivatives of Exponential and Logarithmic Functions
Logarithmic differentiation
Used when the variable is in the base and the exponent
y = xx
ln y = ln xx
ln y = x ln x
xx
xdx
dy
yln
11
xydx
dyln1
xxdx
dy x ln1
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