DT.01.1 - Derivative Rules - Power Rule, Constant Rule, Sum and Difference Rule
Calculus 2.1: Differentiation Formulas A. Derivative of a Constant: B. The Power Rule: C. Constant...
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Transcript of Calculus 2.1: Differentiation Formulas A. Derivative of a Constant: B. The Power Rule: C. Constant...
Calculus 2.1: Differentiation Formulas
d
dxx nxn n( ) 1
d
dxc( ) 0A. Derivative of a
Constant:
B. The Power Rule:
C. Constant Multiple Rule:
d
dxcf x c
d
dxf x( ) ( )
D. Sum Rule:d
dxf x g x
d
dxf x
d
dxg x( ) ( ) ( ) ( )
E. Difference Rule:
d
dxf x g x
d
dxf x
d
dxg x( ) ( ) ( ) ( )
d
dxf x g x f x
d
dxg x g x
d
dxf x[ ( ) ( )] ( ) ( ) ( ) ( )
( )fg f g g f
a f x x x) ( ) ( )( ) 6 73 4
F. Product Rule:
Ex 1: Find f ‘(x):
b y x x) ( )( ) 2 5 1
G. Quotient Rule
fg
gf fg
g
2
( ) ( ) ( ) ( )lo d hi hi d lo
lolo
Ex 2: Find y’
a yx x
x)
2
3
2
6b y
x
x)
1 2
d
dx
g x f x f x g x
g xf xg x
ddx
ddx( )
( )
( ) ( ) ( ) ( )
[ ( )]
2
A. Higher Derivatives
1. Second Derivative: f’’ = (f’)’
(if f’ is differentiable) yd
dx
dy
dx
d y
dx
2
2
2. Third Derivative: f’’’ = (f’’)’(if f’’ is differentiable)
yd
dx
d y
dx
d y
dx
2
2
3
3
Note:y
d
dxy
d y
dxn n
n
n( ) ( ) 1 “y super n”
means the nth derivative
Calculus 2.2: Differentiation Problems
B. Examples
1. Find the equation of a tangent line at the point (1, ½) to the curve:
2. Find the points on the curve y = x4 – 6x2 + 4 where the tangent line is horizontal
3. At what points on the hyperbola xy = 12 is the tangent line parallel to 3x + y = 0?
4. If h(x) = xg(x) and it is known that g(3) = 5 and g‘(3) = 2, find h‘(3)
5. #37 p.124
y x x ( )1 2
Calculus 2.3: More Rates of Change
A. Average Rate of Change of y with respect to x:
y
x
f x f x
x x
( ) ( )2 1
2 1
B. Instantaneous Rate of Change: (derivative)
dy
dx
y
xx
lim
0
C. Applications
1. Linear Motion:a) velocity – the derivative
of the position function s = f(t)
b) speed – the absolute value of velocity
c) acceleration – the derivative of the velocity with respect to time
d) jerk – the derivative of acceleration
v tds
dt( )
speed v t ( )
a tdv
dt
d s
dt( )
2
2
j tda
dt
d s
dt( )
3
3
2. Economics:
a. marginal cost – the rate of change of cost with respect to level of production
b. marginal revenue – the derivative of the revenue function
dc
dx
dr
dx
Calculus 2.4: Derivatives of Trigonometric Functions
d
dxx x(sin ) cos
d
dxx x(cos ) sin
d
dxx x(tan ) sec 2
A. Derivatives of Trig Functions: (Memorize!!)
d
dxx x(cot ) csc 2
d
dxx x x(csc ) csc cotd
dxx x x(sec ) sec tan
Graphing Sin/Cos Functions
Graphing Cos/-Sin Functions
Calculus 2.5: The Chain Rule
dy
dx
dy
du
du
dx
A. The Chain Rule:
If f and g are both differentiable and F = f o g, then F is differentiable and F’ = f ‘(g(x))g‘(x)
Ex 1:
F x x( ) 2 1
B. Power Rule Combined with Chain Rule
d
dxg x n g x g x
n n( ) ( ) ( ) 1
or
d
dxu nu
du
dxn n 1
C. The Chain Rule (Trigonometric Functions)
1 2. siny x
2 2. siny x3. ( ) sin(cos(tan ))f x x
4. secy x
5 5. siny x at x 3
Find tangent line
Calculus 2.6: Implicit Differentiation
dy
dx
1 12 2.x y
2 63 3.x y xy
A. Method of Implicit Differentiation:
1. Differentiate both sides of the equation with respect to x
2. Solve the resulting equation for
B. Examples—find y’ 3
6 422
.xy
x y
4 12. y x
B. Finding
1. Use implicit differentiation to find
2. Differentiate
3. Substitute the expression for
4. Simplify
dy
dx
d y
dx
2
2
d y
dx
2
2
dy
dxdy
dx
C. Implicit Differentiation with Trigonometric Functions
1. sin(x+y) = y2 cos x2. tan(x/y) = x + y3. 4 cos x sin y = 14. 2y = x2 + sin y
Calculus Unit 2 Test
Grademaster #1-30 (Name, Date, Subject, Period, Test Copy #)
Do Not Write on Test! Show All Work on Scratch Paper!
Label BONUS QUESTIONS Clearly on Notebook Paper. (If you have time)
Find Something QUIET To Do When Finished!