Derivative Rules

The Derivative tells you the slope of a function at any point.

The derivatives of many functions are well known. Here are some useful rules to help you work out the derivatives of more complicated functions (with examples below). Note: the little mark means "Derivative of".

Common Functions Function
Derivative
Constant c 0
  x 1
Square x2 2x
Square Root √x (½)x
Exponential ex ex
  ax ax(ln a)
Logarithms ln(x) 1/x
  loga(x) 1 / (x ln(a))
Trigonometry (x is in radians) sin(x) cos(x)
  cos(x) -sin(x)
  tan(x) sec2(x)
  sin-1(x) 1/√(1-x2)
  tan-1(x) 1/(1+x2)
     
Rules Function
Derivative
Multiplication by constant cf cf’
Power Rule xn nxn-1
Sum Rule f + g f’ + g’
Difference Rule f - g f’ - g’
Product Rule fg f g’ + f’ g
Quotient Rule f/g (f’ g - g’ f )/g2
Reciprocal Rule 1/f -f’/f2
Chain Rule
(as "Composition of Functions")
f º g (f’ º g) × g’
Chain Rule (in a different form) f(g(x)) f’(g(x))g’(x)

"The derivative of" is also written

Examples

Example: what is the derivative of sin(x) ?

From the table above it is listed as being cos(x)

It can be written as:

sin(x) = cos(x)

Or:

sin(x)’ = cos(x)

Power Rule

Example: What is x3 ?

The question is asking "what is the derivative of x3?"

We can use the Power Rule, where n=3:

xn = nxn-1

x3 = 3x3-1 = 3x2

Example: What is (1/x) ?

1/x is also x-1

We can use the Power Rule, where n=-1:

xn = nxn-1

x-1 = -1x-1-1 = -x-2

Multiplication by constant

Example: What is 5x3 ?

the derivative of cf = cf’

the derivative of 5f = 5f’

We know (from the Power Rule):

x3 = 3x3-1 = 3x2

So:

5x3 = 5x3 = 5 × 3x2 = 15x2

Sum Rule

Example: What is the derivative of x2+x3 ?

The Sum Rule says:

the derivative of f + g = f’ + g’

So we can work out each derivative separately and then add them.

Using the Power Rule:

  • x2 = 2x
  • x3 = 3x2

And so:

the derivative of x2 + x3 = 2x + 3x2

Difference Rule

It doesn't have to be x, we can differentiate with respect to, for example, v:

Example: What is (v3-v4) ?

The Difference Rule says

the derivative of f - g = f’ - g’

So we can work out each derivative separately and then subtract them.

Using the Power Rule:

  • v3 = 3v2
  • v4 = 4v3

And so:

the derivative of v3 - v4 = 3v2 - 4v3

Sum, Difference, Constant Multiplication And Power Rules

Example: What is (5z2 + z3 - 7z4) ?

Using the Power Rule:

  • z2 = 2z
  • z3 = 3z2
  • z4 = 4z3

And so:

(5z2 + z3 - 7z4) = 5 × 2z + 3z2 - 7 × 4z3 = 10z + 3z2 - 28z3

 

Product Rule

Example: What is the derivative of cos(x)sin(x) ?

The Product Rule says:

the derivative of fg = f g’ + f’ g

In our case:

  • f = cos
  • g = sin

We know (from the table above):

  • cos(x) = -sin(x)
  • sin(x) = cos(x)

So:

the derivative of cos(x)sin(x) = cos(x)cos(x) - sin(x)sin(x)

= cos2(x) - sin2(x)

 

Reciprocal Rule

Example: What is (1/x) ?

The Reciprocal Rule says:

the derivative of 1/f = -f’/f2

With f(x)= x, we know that f’(x) = 1

So:

the derivative of 1/x = -1/x2

Which is the same result we got above using the Power Rule.

Chain Rule

Example: What is sin(x2) ?

sin(x2) is made up of sin() and x2:

  • f(g) = sin(g)
  • g(x) = x2

The Chain Rule says:

the derivative of f(g(x)) = f'(g(x))g'(x)

The individual derivatives are:

  • f'(g) = cos(g)
  • g'(x) = 2x

So:

sin(x2) = cos(g(x)) × 2x = 2x cos(x2)

 

Example: What is (1/sin(x)) ?

1/sin(x) is made up of 1/g and sin():

  • f(g) = 1/g
  • g(x) = sin(x)

The Chain Rule says:

the derivative of f(g(x)) = f’(g(x))g’(x)

The individual derivatives are:

  • f'(g) = -1/(g2)
  • g'(x) = cos(x)

So:

(1/sin(x))’ = -1/(g(x))2 × cos(x) = -cos(x)/sin2(x)

 

Example: What is (5x-2)3 ?

The Chain Rule says:

the derivative of f(g(x)) = f’(g(x))g’(x)

(5x-2)3 is made up of g3 and 5x-2:

  • f(g) = g3
  • g(x) = 5x-2

The individual derivatives are:

  • f'(g) = 3g2 (by the Power Rule)
  • g'(x) = 5

So:

(5x-2)3 = 3g(x)2 × 5 = 15(5x-2)2