Chapter 5 Higher Derivatives
Mindmap of C5
See Patterns
Higher Derivatives
Partial Fractions
Direct Evaluation
Leibniz's Rule
Differential Equations of High Orders
C5
Derivatives
Asymptotes to a Curve
Points of Inflexion
Either Decreasing or Increasing
Extrema
Inequatities
Decreasing and Increasing
Inequalities
Greatest & Least Values
Curve Sketching
Convexity of a Function
Definition
dy
f (x  h)  f (x)
First derivative  lim
,
dx h 0
h
d 2 y d  dy 
Second derivative 2 
 ,
dx
dx  dx 
...
...
d n y d  d n -1 y 
 n 1 .
Inductively, n 
dx
dx  dx 
They are also denoted by y(1), y(2),…, y(n). y(0) denotes y.
Find the 10th derivatives of the
following functions
xm
sinx
1/x
1/(ax+b)
lnx
eax
1/(x-4)(x+2)
Find the nth derivatives of the
following functions
xm
sinx
1/x
1/(ax+b)
lnx
eax
1/(x-4)(x+2)
If n<=m, (xm)(n) =m(m-1)…(mn+1)xm-n. If n>m, (xm)(n)= 0.
(-1)nn!/xn+1
(-1)nann!/(ax+b)n+1
(-1)n-1(n-1)!/xn
aneax
Can you give formal
proofs of your results?
More nth derivatives
1/(x2-2x-3)
xlnx
sinx
exsinx
¼ (1/(x-3))(n) – ¼ (1/(x+1))(n)
¼(-1)nn![1/(x-3)n+1 – 1/(x+1)n+1]
y’=1+lnx
y(n)=(-1)n(n -2)!/xn-1 for n>=2.
y’=cosx, y”=-sinx, y”’=-cosx, y(4)=sinx
y(n)=sin(n/2+x)
Difficult. See p.166
Questions
(x+ex)(10)=?
(f + g)(n)=f(n) + g(n)
(xex)(10)=?
(fg)(n)=f(n).g(n) ?
No! So how?
Show time
(x2ex)(5)=?
(fg)(n)
Find (fg)(n) for n = 1,2,3,4,5.
Guess the formula for (fg)(n).
Leibniz’s Rule
n
(fg)
(n)
 C f
r 0
Proof :
n ( n r )
r
g
(r)
Example 1
Let y = x2sinx, find y(80).
Example 2
If y = x3eax, find dny/dxn.
Example 3
Find the nth derivative of y = 2xlnx.
Example 4
Let y be function of x, which is differentiable any
times. If (1 – x2)y” – xy’ – a2y = 0, show that for
any positive integer n,
(1 – x2)y(n+2) – (2n+1)xy(n+1) – (n2 + a2)y(n) = 0.
CW Ex.5.2, 7
Section 4 Extrema and inequalities
Can you write down an
inequality?
Can you write down an
inequality?
y=ln(1+x) – x + x2/2
y
y
x
x
y=ln(1+x) – x + x2/2 – x3/3
Can you combine these two inequalities?
Example 4.4
If x > 0, prove, without using graphs, that
x – x2/2 < ln(1+x) < x – x2/2 + x3/3!
Steps:
Consider h(x) = ln(1+x) – x +x2/2
Evaluate h’(x).
Consider whether it’s monotonic.
Conclusion
Inequalities by monotonicity
Theorem: If f’(x) > g’(x) for x > a and f(a) = g(a), then f(x) < g(x).
Steps :
1. Let h(x) = f(x) – g(x).
2. Then h’(x) = f’(x) – g’(x).
3. h’(x) > 0(< 0) , then h(x) is strictly increasing(decreasing).
4. h(x) > h(a)= 0(< h(a) for all x > a.
5. f(x) – g(x) > 0 for all x > a
6. f(x) > g(x) for all x. > a
(Q.E.D.)
Section 6 Greatest and Least Values
e.g.6.2
Show that x – lnx >= 1 for all x > 0.
Proof:
Let f(x) = x – lnx , then
f’(x) = 1 – 1/x > 0 for x > 1 and
< 0 for 0< x < 1,
f(1) is the least value of f.
f(x)>=f(1)=1
i.e. x – lnx >=1
When does the equal sign hold?
y
x
Section 8 Convexity of a Function
Definition 8.3 The following graph of a realvalued function f(x) is said to be convex
downward.
Find a condition for it.
1. f ”(x)
y
x
Section 8 Convexity of a Function
Definition 8.4 The following graph of a realvalued function f(x) is said to be convex upward.
Find a condition for it.
y
x