Derivatives (D):
•
•
•
d
dy
f (x ) ⇒ f ′( x ) ⇒
dx
dx
d
the derivative.
dx
The derivative represents the slope of the function at some x, and slope represents a rate of
change at that point.
The derivative (Dx) of a constant (C) is zero.
In this tutorial I will use Dx for
Power Rule: The fundamental tool for finding the Dx of f(x).
d n
x ⇒ f ′( x) ⇒ nx n −1 dx *
dx
Ex:
Dx [x ] ⇒ f ′( x ) = 3x 2 dx = 3x 2
3
3
− multiply the exponent times the coefficient of x and then reduce the exponent by 1.
* [dx represents the derivative of what is inside (x), which is usually 1 for simple functions, the dx must
always be considered and is always there, even if it is only 1]
Sum Rule:
The Dx of a sum is equal to the sum of the Dx’s
d
[ f (x ) + g (x )] ⇒ f ' (x ) + g ' (x )
dx
d
Ex:
3x 2 + 2 x + 3 ⇒ f ′ 3x 2 + f ' (2 x) + f ' (3) ⇒ 6 x + 2 + zero
dx
[
]
( )
Constant Coefficient Rule: The Dx of a variable with a constant coefficient is equal to the constant
times the Dx. The constant can be initially factored out.
[ ]
d
Cx n ⇒ C f ′( x ) ⇒ Cnx n −1 dx
dx
Ex:
(
)
( )
d
d 2
ln(4) x 2 = ln(4) [
x ] = ln(4) [2x] dx = 2ln(4)x = ln(16)x
dx
dx
Product Rule:
other function.
The Dx of a product is equal to the sum of the products of the Dx’s times the
d
[ f ( x) • g ( x)] ⇒ [ f ′(x ) dx • g ( x)] + [ f ( x) • g ′( x) dx]
dx
[
] [ ( ) ] [
] [
] [
]
d
3x 2 y 3 ⇒ f ′ 3 x 2 y 3 + 3x 2 g ' ( y 3 ) = 6 x dx • y 3 + 3x 2 • 3 y 2 dy = 6 xy 3 + 9 x 2 y 2 dy *
dx
* Note: when y is differentiated with respect to x, there will be a “dy” left in the result (see implicit
differentiation).
James Sebring 01/01/084
Chain Rule: There is nothing new here other than the dx is now something other than 1. The dx represents
the Dx of the inside function g(x). It is called a chain rule because you have to consider the dx as not being 1
and take the Dx of the inside also.
d
[ f ( g ( x))]⇒ f ′(g ( x)) ⇒ f ′( g ( x)) dx ⇒ f ′( g ( x)) g ′( x)
dx
* [the dx here is g’(x)]
Ex:
Dx (sin(3x)) = cos(3x) dx* = 3 cos(3x)
* [dx is g’(3x) = 3]
Dx ((3x2+2)2 = 2(3x2+2 ) dx* = 2(3x2+2 ) (6x) = (6x2 + 4)(6x) = 36x3 + 24x
*[dx is Dx (3x2 + 2) = 6x] notice we used the Power Rule along with the Chain Rule
Quotient Rule:
Dx (numerator) times the denominator minus Dx (denominator) times the
numerator, divided by the denominator squared. This is a variation of the Product Rule.
d ⎡ f ( x) ⎤
f ' ( x) g ( x) − f ( x) g ' ( x)
⇒
⎢
⎥
dx ⎣ g ( x) ⎦
[ g ( x)]2
Ex:
d ⎡ sin( x) ⎤
cos( x)(3 x) − sin( x)(3) 3 x cos( x) − 3 sin( x) x cos( x) − sin( x)
⇒
=
=
⎢
⎥
dx ⎣ 3 x ⎦
[3 x] 2
9x 2
3x 2
Special Rules:
d ⎡1⎤
⇒ ln( x) dx
dx ⎢⎣ x ⎥⎦
d
[log b ( x)] ⇒ 1 dx
dx
x ln(b)
1
1
d
[ln(sin x)] ⇒
cos x = cot x
dx =
sin x ln(e)
sin x
dx
[ ]
d
y = 3 x ⇒ [3 x ] ln(3) dx ⇒ 3 x ln(3)
dx
[
]
d 4 x=2
e
= e 4 x = 2 dx = 4e 4 x = 2
dx
Substitution:
Applications:
Physics:
Dx (position or distance) = velocity; Dx (velocity) = acceleration
Sf = at2 + vt + S0
Vf = at + v0
James Sebring 01/01/084