Derivatives

Definition and Notation

f ( x + h) - f ( x)

.

If y = f ( x ) then the derivative is defined to be f ¢ ( x ) = lim h ®0

h

If y = f ( x ) then all of the following are

equivalent notations for the derivative.

df dy d

f ¢ ( x ) = y¢ =

=

= ( f ( x ) ) = Df ( x )

dx dx dx

If y = f ( x ) then,

If y = f ( x ) all of the following are equivalent

notations for derivative evaluated at x = a .

df

dy

f ¢ ( a ) = y ¢ x =a =

=

= Df ( a )

dx x =a dx x =a

Interpretation of the Derivative

2. f ¢ ( a ) is the instantaneous rate of

1. m = f ¢ ( a ) is the slope of the tangent

change of f ( x ) at x = a .

line to y = f ( x ) at x = a and the

3. If f ( x ) is the position of an object at

time x then f ¢ ( a ) is the velocity of

equation of the tangent line at x = a is

given by y = f ( a ) + f ¢ ( a ) ( x - a ) .

the object at x = a .

Basic Properties and Formulas

If f ( x ) and g ( x ) are differentiable functions (the derivative exists), c and n are any real numbers, 1.

( c f )¢ = c f ¢ ( x )

2.

( f ± g )¢ = f ¢ ( x ) ± g ¢ ( x )

3.

( f g )¢ =

æf

4. ç

èg

d

(c) = 0

dx

dn

6.

( x ) = n xn-1 – Power Rule

dx

d

7.

f ( g ( x )) = f ¢ ( g ( x )) g¢ ( x )

dx

This is the Chain Rule

5.

f ¢ g + f g ¢ – Product Rule

ö¢ f ¢ g - f g ¢

– Quotient Rule

÷=

g2

ø

(

)

Common Derivatives

d

( x) = 1

dx

d

( sin x ) = cos x

dx

d

( cos x ) = - sin x

dx

d

( tan x ) = sec2 x

dx

d

( sec x ) = sec x tan x

dx

d

( csc x ) = - csc x cot x

dx

d

( cot x ) = - csc2 x

dx

d

( sin -1 x ) = 1 2

dx

1- x

d

( cos-1 x ) = - 1 2

dx

1- x

d

1

( tan -1 x ) = 1 + x2

dx

Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.

dx

( a ) = a x ln ( a )

dx

dx

(e ) = ex

dx

d

1

( ln ( x ) ) = x , x > 0

dx

d

( ln x ) = 1 , x ¹ 0

dx

x

d

1

( log a ( x ) ) = x ln a , x > 0

dx

© 2005 Paul Dawkins

Calculus Cheat Sheet

Chain Rule Variants

The chain rule applied to some specific functions.

n

n -1

d

d

1.

é f ( x )ù = n é f ( x )ù f ¢ ( x )

5.

cos é f ( x ) ù = - f ¢ ( x ) sin é f ( x ) ù

ë

û

ë

û

ë

û

ë

û

dx

dx

d f ( x)

d

fx

e

tan é f ( x ) ù = f ¢ ( x ) sec 2 é f ( x ) ù

2.

= f ¢( x)e ( )

6.

ë

û

ë

û

dx

dx

d

f ¢( x)

d

7.

( sec [ f ( x)]) = f ¢( x) sec [ f ( x)] tan [ f ( x)]

3.

ln é f ( x ) ù =

ë

û

dx

dx

f ( x)

f ¢( x)

d

d

tan -1 é f ( x ) ù =

8.

2

4.

sin é f ( x ) ù = f ¢ ( x ) cos é f ( x ) ù

ë

û

ë

û

ë

û

dx

1 + é f ( x )ù

dx

ë

û

)

(

(

(

(

)

(

)

)

)

(

(

)

)

Higher Order Derivatives

The Second Derivative is denoted as

The nth Derivative is denoted as

d2 f

dn f

f ¢¢ ( x ) = f ( 2) ( x ) = 2 and is defined as

f ( n ) ( x ) = n and is defined as

dx

dx

¢

f ¢¢ ( x ) = ( f ¢ ( x ) )¢ , i.e. the derivative of the f ( n ) ( x ) = f ( n -1) ( x ) , i.e. the derivative of

first derivative, f ¢ ( x ) .

the (n-1)st derivative, f ( n-1) x .

(

)

()

Implicit Differentiation

¢ if e 2 x -9 y + x3 y 2 = sin ( y ) + 11x . Remember y = y ( x ) here, so products/quotients of x and y Find y

will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y¢ (from the chain rule). After differentiating solve for y¢ .

e 2 x -9 y ( 2 - 9 y¢ ) + 3 x 2 y 2 + 2 x3 y y¢ = cos ( y ) y¢ + 11 2e

2 x -9 y

- 9 y¢e

( 2 x y - 9e x

3

2 x -9 y

2 -9 y

+ 3x y + 2 x y y¢ = cos ( y ) y¢ + 11

2

2

3

- cos ( y ) ) y¢ = 11 - 2e2 x -9 y - 3x 2 y 2

Þ

11 - 2e 2 x -9 y - 3x 2 y 2

y¢ = 3

2 x y - 9e2 x -9 y - cos ( y )

Increasing/Decreasing – Concave Up/Concave Down

Critical Points

x = c is a critical point of f ( x ) provided either

1. f ¢ ( c ) = 0 or 2. f ¢ ( c ) doesn’t exist.

Increasing/Decreasing

1. If f ¢ ( x ) > 0 for all x in an interval I then...