background mathematics review david miller · dx 2 2 0 dy dx xx xxx. sign of second derivative...
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Differential calculus
Background mathematics review David Miller
Differential calculus
First derivative
Background mathematics review David Miller
First derivative
For some functionThe (first) derivative is the
“slope” gradient orrate of change
of y as we change xIf for some small “infinitesimal”
change in x, called dxy changes by some small “infinitesimal” amount dy
y
x
y x
the first derivative is written
The “ratio” notation on the right is “Leibniz notation”
dyy xdx
y
xx1
1x
d yd x
First derivative
The derivative at some specific point x1 can be written
The value of the derivative is the slope of the “tangent” line
the dashed line in the figureat that point
Equal in value to the tangent
1
1x
dyy xdx
x3
3x
d yd x
xy
yx
First derivative
Looking at the slope
of the “orange” lineas we reduce
the “orange” line slope becomes closer to the slope of the “black” tangent line
y
x
x
x1
1 2xy x
1 2xy x
1 12 2
x xy x y x
x
x
First derivative
In the limit as becomes very smalli.e., in the limit as “tends to zero”
this ratio becomes the (first) derivative
xx
0limx
1
1 1
0
2 2limx
x
x xy x y xdydx x
Sign of first derivative
If y increases as we increase x
sloping up to the right
If y decreases as we increase x
sloping down to the right
y
xx1
0d yd x
0d yd x
x3
0d yd x
0d yd x
Derivative of a power
1 0.5 0 0.5 1
0.5
12y x
1n nd x nxdx
2dy x
dx
1 0.5 0 0.5 1
2
2
Derivative of a power
The derivative of a straight line The straight line has a constant slope
is a constant
1 0.5 0 0.5 1
1
1 y x
1 0.5 0 0.5 1
0.5
11dy
dx
Derivative of a power
The derivative does not depend on the “height” All these lines have the same slope
1 0.5 0 0.5 1
1
1y x
1 0.5 0 0.5 1
0.5
11dy
dx
1 0.5 0 0.5 1
1
1
Derivative of a power
The derivative does not depend on the “height” All these lines have the same slope
0.25y x
1 0.5 0 0.5 1
0.5
11dy
dx
1 0.5 0 0.5 1
1
1
Derivative of a power
The derivative does not depend on the “height” All these lines have the same slope
0.5y x
1 0.5 0 0.5 1
0.5
11dy
dx
Derivative of a power
The derivative of a constanty is not changing with x
is zero
1 0.5 0 0.5 1
1
10.5y
1 0.5 0 0.5 1
0.5
1
0dydx
Derivative of an exponential
exp expd x xdx
1 0.5 0 0.5 1
1
2
3
expy x
1 0.5 0 0.5 1
1
2
3
expdy xdx
Derivative of a logarithm
1lnd xdx x
0 1 2 3
3
2
1
1
2 lny x
0 1 2 3
5
10
1dydx x
Derivatives of sine and cosine
sin cosd x xdx
1
1
siny x
1
1
cosdy xdx
Derivatives of sine and cosine
cos sind x xdx
1
1
cosy x
1
1
sindy xdx
Differential calculus
Second derivative
Background mathematics review David Miller
Second derivative
The second derivative isThe derivative of the derivative
The rate of change of the derivative or “slope”
2
2
d y d dyy xdx dx dx
1 0.5 0 0.5 1
0.5
1
x
Second derivative
The slope at is, for small
And similarly at
So 1 0.5 0 0.5 1
0.5
1
x
/ 2x x
/2
0
x
y y xdydx x
/ 2x
/2
0
x
y x ydydx x
2
2 0limx
d y d dydx dx dx
/2 /2x x
dy dydx dx
x
Second derivative
The slope at is, for small
And similarly at
So 1 0.5 0 0.5 1
0.5
1
x
/ 2x x
/2
0
x
y y xdydx x
/ 2x
/2
0
x
y x ydydx x
2
2 0limx
d y d dydx dx dx
0 01 y x y y y xx x x
Second derivative
The slope at is, for small
And similarly at
So 1 0.5 0 0.5 1
0.5
1
x
/ 2x x
/2
0
x
y y xdydx x
/ 2x
/2
0
x
y x ydydx x
2
2 0limx
d y d dydx dx dx
2
2 0y x y y x
x
Sign of second derivative
Going from a positive first derivative
To a negative first derivativeGives a negative second
derivativeGoing from a negative first
derivativeTo a positive first derivative
Gives a positive second derivative
y
xx2 x3 x4x1
0d yd x
2
2 0d ydx
0d y
d x
2
2 0d ydx
Sign of second derivative
Any region where the first derivative is decreasing with increasing x
Has a negative second derivative
Any region where the first derivative is increasing with increasing x
Has a positive second derivative
y
xx
2
2 0d yd x
2
2 0d yd x
x x x x x
Sign of second derivative
Points where the derivative is neither increasing or decreasing
i.e., second derivative is changing sign
correspond to zero second derivative
Known as inflection points
y
xx
2
2 0d yd x
2
2 0d yd x
x
Curvature
The second derivative can be thought of as the
“curvature”of a function
Large positive curvature
Curvature
The second derivative can be thought of as the
“curvature”of a function
Small positive curvature
Curvature
The second derivative can be thought of as the
“curvature”of a function
Large negative curvature
Curvature
The second derivative can be thought of as the
“curvature”of a function
Small negative curvature
Curvature
The value of the curvature does not depend on the “height” of the function
All these curves have the same curvature
Differential calculus
Linearity and differentiation rules
Background mathematics review David Miller
Linearity – linear superposition
For two functions
and
The derivative of the sum is the sum of the derivatives
Example
Split into
So
So
u x v x
d du dvu x v xdx dx dx
lnf x x x
u x x lnv x x
1dudx
1dv
dx x
11d u v
f xdx x
Linearity – multiplying by a constant
For a functionThe derivative of
a constant a times a function
is a times the derivative
Example
Split into
So
So
u x
d duau adx dx
1/2f x a x a x
a 1/2u x x
1/21 12 2
du xdx x
2
du af x adx x
Linearity
An operation or function is linear if
“linear superposition” or “additivity” condition
and
“multiplication by a constant” (or formally “homogeneity of degree one”) condition
f x f y z f y f z
f ax a f x
Example of nonlinear operation
The function
does not represent a linear operation
But
So for this function
is not in general equal to
2f x x
2 2 2 2f y z y z y z yz
2 2f y f z y z
f x y
f x f y
Product rule
For two functions
and
The derivative of the product is
Example
Split into
So
So
u x v x
d dv duuv u vdx dx dx
2 sinf x x x
2u x x sinv x x
2du xdx
cosdv xdx
2 cos 2 sind uv
f x x x x xdx
Quotient rule
For two functions
and
The derivative of the ratio or quotient is
Example
Split into
So
So2
du dvv ud u dx dxdx v v
u x v x 3
21xf x
x
3u x x 21v x x
23du xdx
2dv xdx
d udx v
2 2 3
22
1 3 2
1
x x x x
x
Quotient rule
For two functions
and
The derivative of the ratio or quotient is
Example
Split into
So
So2
du dvv ud u dx dxdx v v
u x v x 3
21xf x
x
3u x x 21v x x
23du xdx
2dv xdx
d udx v
2 2 3
22
3 1 2
1
x x x x
x
Quotient rule
For two functions
and
The derivative of the ratio or quotient is
Example
Split into
So
So2
du dvv ud u dx dxdx v v
u x v x 3
21xf x
x
3u x x 21v x x
23du xdx
2dv xdx
d udx v
2 2 4
22
3 1 2
1
x x x
x
Quotient rule
For two functions
and
The derivative of the ratio or quotient is
Example
Split into
So
So2
du dvv ud u dx dxdx v v
u x v x 3
21xf x
x
3u x x 21v x x
23du xdx
2dv xdx
d udx v
2 4
22 2
3 21 1
x xx x
Chain rule
For two functions
and
The derivative of the “function of a function”
Can be split into a product
Example
Split into
d df dgf g xdx dg dx
f y g x 221h x x
21g x x 2f y y
Chain rule
For two functions
and
The derivative of the “function of a function”
Can be split into a product
Example
Split into
So
So d df dgf g x
dx dg dx
f y g x 221h x x
21g x x 2f g g
2
df gg
dg2dg x
dx
2 22 1 2 4 1dh x x x xdx
Chain rule
For two functions
and
The derivative of the “function of a function”
Can be split into a product
Example
Split into
So
So d df dgf g x
dx dg dx
f y g x exph x a x
g x a x expf g g
expdf g
gdg
dg adx
exp expdh ax a a axdx
Chain rule
For two functions
and
The derivative of the “function of a function”
Can be split into a product
Example
Split into
So
So d df dgf g x
dx dg dx
f y g x 2exph x a x
2g x a x expf g g
expdf g
gdg
2dg axdx
2 2exp 2 2 expdh ax ax ax axdx
