Derivative formulas
September 29, 2013
Derivative formulas
Derivative of a constant function
Derivative is the slope of the graph.
The graph of a constant function is a horizontal line with the
slope 0 everywhere.
Derivative formulas
Derivative of a constant function
Derivative is the slope of the graph.
The graph of a constant function is a horizontal line with the
slope 0 everywhere.
If f (x) = k, then f 0 (x) = 0.
Derivative formulas
Derivative of a constant function
Derivative is the slope of the graph.
The graph of a constant function is a horizontal line with the
slope 0 everywhere.
If f (x) = k, then f 0 (x) = 0.
d
d
Example
(9) =
(3) = 0.
dx
dx
Derivative formulas
Derivative of a linear function
Derivative is the slope of the graph.
The graph of a linear function is a line, with a fixed slope.
Derivative formulas
Derivative of a linear function
Derivative is the slope of the graph.
The graph of a linear function is a line, with a fixed slope.
If f (x) = mx + b, then f 0 (x) = m.
Derivative formulas
Derivative of a linear function
Derivative is the slope of the graph.
The graph of a linear function is a line, with a fixed slope.
If f (x) = mx + b, then f 0 (x) = m.
d
(4x + 6) = 4.
Example
dx
Derivative formulas
Linear properties of derivative
Theorem
Let f and g be two functions, and a and b two constants.
d
[cf (x)] = cf 0 (x)
dx
Derivative formulas
Linear properties of derivative
Theorem
Let f and g be two functions, and a and b two constants.
d
[cf (x)] = cf 0 (x)
dx
d
[f (x) + g (x)] = f 0 (x) + g 0 (x)
dx
Derivative formulas
Linear properties of derivative
Theorem
Let f and g be two functions, and a and b two constants.
d
[cf (x)] = cf 0 (x)
dx
d
[f (x) + g (x)] = f 0 (x) + g 0 (x)
dx
d
[f (x) − g (x)] = f 0 (x) − g 0 (x)
dx
Derivative formulas
Linear properties of derivative
Theorem
Let f and g be two functions, and a and b two constants.
d
[cf (x)] = cf 0 (x)
dx
d
[f (x) + g (x)] = f 0 (x) + g 0 (x)
dx
d
[f (x) − g (x)] = f 0 (x) − g 0 (x)
dx
d
[af (x) + bg (x)] = af 0 (x) + bg 0 (x)
dx
Derivative formulas
Power rule
For any constant real number n,
d n
(x ) = nx n−1 .
dx
Derivative formulas
Example
Find the derivative of the following functions:
Derivative formulas
Example
Find the derivative of the following functions:
1
f (x) = √ .
x
Derivative formulas
Example
Find the derivative of the following functions:
1
f (x) = √ .
x
2
t
P(t) = √
.
3
t
Derivative formulas
Example
Find the derivative of the following functions:
1
f (x) = √ .
x
2
t
P(t) = √
.
3
t
g (w ) = w 4 + 6.
Derivative formulas
Example
Find the derivative of the following functions:
1
f (x) = √ .
x
2
t
P(t) = √
.
3
t
g (w ) = w 4 + 6.
Q(x) = x 4 + 3x 2 − x + 1.
Derivative formulas
Example
Find the derivative of the following functions:
1
f (x) = √ .
x
2
t
P(t) = √
.
3
t
g (w ) = w 4 + 6.
Q(x) = x 4 + 3x 2 − x + 1.
p
h(θ) = θ2 ( (θ) + 4θ−6 ).
Derivative formulas
Using the derivative formulas
Find the tangent line at x = 2 to the graph of
y = x 4 + 2x 2 + 6.
Derivative formulas
Using the derivative formulas
Find the tangent line at x = 2 to the graph of
y = x 4 + 2x 2 + 6.
Slope of the line=?
Derivative formulas
Using the derivative formulas
Find the tangent line at x = 2 to the graph of
y = x 4 + 2x 2 + 6.
Slope of the line=?
Given point on the line=?
Derivative formulas
Example
Find and interpret the second derivative of f (x) = x 3 + 12x.
Derivative formulas
Example
The revenue (in dollars) from producing q units of a product is
given by
R(q) = 2500q − 5q 2
Find and interpret R(100) and R 0 (100).
Derivative formulas
Exponential rule
For any positive constant a
d
x
dx (a )
= (ln a)ax
Derivative formulas
Exponential rule
For any positive constant a
d
x
x
dx (a ) = (ln a)a
d
kx
kx
dx (a ) = k(ln a)a
Derivative formulas
Exponential rule
For any positive constant a
d
x
x
dx (a ) = (ln a)a
d
kx
kx
dx (a ) = k(ln a)a
d
kx
kx
dx (e ) = ke
Derivative formulas
Examples
Find the derivative of
y = 5 + 4x 2 + e −0.02x .
Derivative formulas
Examples
Find the derivative of
y = 5 + 4x 2 + e −0.02x .
√
P = x + 3 · 5x + 4e −4x
Derivative formulas
Derivative of ln x
Theorem
d
1
(ln x) =
dx
x
Derivative formulas
Examples
Find the derivative of y = 5 ln x 4 + 6x + 4e −2x .
Derivative formulas
Examples
Suppose $1000 is deposited into a bank account that pays 8%
annual interest, compounded continuously.
Find f (10) and f 0 (10).
Derivative formulas
Examples
Suppose $1000 is deposited into a bank account that pays 8%
annual interest, compounded continuously.
Find f (10) and f 0 (10).
Explain what your answers mean in terms of money.
Derivative formulas