Derivative Rules

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

slope examples y=3, slope=0; y=2x, slope=2

There are rules we can follow to find many derivatives.

For example:

Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark means "Derivative of", and f and g are functions.

Common FunctionsFunction
Derivative
Constantc0
Linex1
 axa
Squarex22x
Square Root√x(½)x
Exponentialexex
 axln(a) ax
Logarithmsln(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)
Inverse Trigonometrysin-1(x)1/√(1−x2)
 cos-1(x)−1/√(1−x2)
 tan-1(x)1/(1+x2)
   
RulesFunction
Derivative
Multiplication by constantcfcf’
Power Rulexnnxn−1
Sum Rulef + gf’ + g’
Difference Rulef - gf’ − g’
Product Rulefgf g’ + f’ g
Quotient Rulef/g(f’ g − g’ f )/g2
Reciprocal Rule1/f−f’/f2
   
Chain Rule
(as "Composition of Functions")
f º g(f’ º g) × g’
Chain Rule (using ’ )f(g(x))f’(g(x))g’(x)
Chain Rule (using ddx )dydx = dydududx

"The derivative of" is also written ddx

So ddxsin(x) and sin(x)’ both mean "The derivative of sin(x)"

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:

d/dxsin(x) = cos(x)

Or:

sin(x)’ = cos(x)

Power Rule

Example: What is d/dxx3 ?

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

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

d/dxxn = nxn−1

d/dxx3 = 3x3−1 = 3x2

(In other words the derivative of x3 is 3x2)

So it is simply this:

power rule x^3 -> 3x^2
"multiply by power
then reduce power by 1"

It can also be used in cases like this:

Example: What is d/dx(1/x) ?

1/x is also x-1

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

d/dxxn = nxn−1

d/dxx−1 = −1x−1−1

= −x−2

= −1x2

So we just did this:

power rule x^-1 -> -x^-2
which simplifies to −1/x2

Multiplication by constant

Example: What is d/dx5x3 ?

the derivative of cf = cf’

the derivative of 5f = 5f’

We know (from the Power Rule):

d/dxx3 = 3x3−1 = 3x2

So:

d/dx5x3 = 5d/dxx3 = 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:

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 d/dv(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:

And so:

the derivative of v3 − v4 = 3v2 − 4v3

Sum, Difference, Constant Multiplication And Power Rules

Example: What is d/dz(5z2 + z3 − 7z4) ?

Using the Power Rule:

And so:

d/dz(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:

We know (from the table above):

So:

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

= cos2(x) − sin2(x)

 

Quotient Rule

To help you remember:

(fg)’ = gf’ − fg’g2

The derivative of "High over Low" is:

"Low dHigh minus High dLow, over the line and square the Low"

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

In our case:

We know (from the table above):

So:

the derivative of cos(x)x = Low dHigh minus High dLowover the line and square the Low

= x(−sin(x)) − cos(x)(1)x2

= −xsin(x) + cos(x)x2

 

Reciprocal Rule

Example: What is d/dx(1/x) ?

The Reciprocal Rule says:

the derivative of 1f = −f’f2

With f(x)= x, we know that f’(x) = 1

So:

the derivative of 1x = −1x2

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

Chain Rule

Example: What is ddxsin(x2) ?

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

The Chain Rule says:

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

The individual derivatives are:

So:

ddxsin(x2) = cos(g(x)) (2x)

= 2x cos(x2)

Another way of writing the Chain Rule is: dydx = dydududx

Let's do the previous example again using that formula:

Example: What is ddxsin(x2) ?

dydx = dydududx

Have u = x2, so y = sin(u):

ddx sin(x2) = ddusin(u)ddxx2

Differentiate each:

ddx sin(x2) = cos(u) (2x)

Substitue back u = x2 and simplify:

ddx sin(x2) = 2x cos(x2)

Same result as before (thank goodness!)

Another couple of examples of the Chain Rule:

Example: What is d/dx(1/cos(x)) ?

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

The Chain Rule says:

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

The individual derivatives are:

So:

(1/cos(x))’ = −1/(g(x))2 × −sin(x)

= sin(x)/cos2(x)

Note: sin(x)/cos2(x) is also tan(x)/cos(x), or many other forms.

 

Example: What is d/dx(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:

The individual derivatives are:

So:

d/dx(5x−2)3 = 3g(x)2 × 5 = 15(5x−2)2