Derivative Rules - Math Is Fun

Derivative Rules

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

There are rules we can follow to find many derivatives.

For example:

• The slope of a constant value (like 3) is always 0
• The slope of a line like 2x is 2, or 3x is 3 etc
• and so on.

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 Functions	Function	Derivative
Constant	c	0
Line	x	1
ax	a
Square	x2	2x
Square Root	√x	(½)x-½
Exponential	ex	ex
ax	ln(a) ax
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)
Inverse Trigonometry	sin-1(x)	1/√(1−x2)
cos-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’ fg2
Reciprocal Rule	1/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 d dx )	dy dx = dy du du dx

"The derivative of" is also written d dx

So d dx sin(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:

ddxsin(x) = cos(x)

Or:

sin(x)’ = cos(x)

Power Rule

Example: What is ddxx3 ?

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

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

ddxxn = nxn−1

ddxx3 = 3x3−1 = 3x2

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

So it is simply this:

"multiply by power then reduce power by 1"

It can also be used in cases like this:

Example: What is ddx(1/x) ?

1/x is also x-1

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

ddxxn = nxn−1

ddxx-1 = −1x-1−1

= −x-2

= −1x2

So we just did this:

which simplifies to −1/x2

Multiplication by constant

Example: What is ddx5x3 ?

the derivative of cf = cf’

the derivative of 5f = 5f’

We know (from the Power Rule):

ddxx3 = 3x3−1 = 3x2

So:

ddx5x3 = 5ddxx3 = 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:

• ddxx2 = 2x
• ddxx3 = 3x2

And so:

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

Difference Rule

What we differentiate with respect to doesn't have to be x, it could be anything. In this case v:

Example: What is ddv(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:

• ddvv3 = 3v2
• ddvv4 = 4v3

And so:

the derivative of v3 − v4 = 3v2 − 4v3

Sum, Difference, Constant Multiplication And Power Rules

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

Using the Power Rule:

• ddzz2 = 2z
• ddzz3 = 3z2
• ddzz4 = 4z3

And so:

ddz(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):

• ddxcos(x) = −sin(x)
• ddxsin(x) = cos(x)

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:

• f = cos
• g = x

We know (from the table above):

• f' = −sin(x)
• g' = 1

So:

the derivative of cos(x)x = Low dHigh minus High dLowsquare the Low

= x(−sin(x)) − cos(x)(1)x2

= −xsin(x) + cos(x)x2

Reciprocal Rule

Example: What is ddx(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 d dx 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:

d dx sin(x2) = cos(g(x)) (2x)

= 2x cos(x2)

Another way of writing the Chain Rule is: dy dx = dy du du dx

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

Example: What is d dx sin(x2) ?

dy dx = dy du du dx

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

d dx sin(x2) = d du sin(u) d dx x2

Differentiate each:

d dx sin(x2) = cos(u) (2x)

Substitute back u = x2 and simplify:

d dx sin(x2) = 2x cos(x2)

Same result as before (thank goodness!)

Another couple of examples of the Chain Rule:

Example: What is ddx(1/cos(x)) ?

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

• f(g) = 1/g
• g(x) = cos(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) = −sin(x)

So:

(1/cos(x))’ = −1g(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 ddx(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:

ddx(5x−2)3 = (3g(x)2)(5) = 15(5x−2)2

6800, 6801, 6802, 6803, 6804, 6805, 6806, 6807, 6808, 6809, 6810, 6811, 6812 Introduction to Derivatives Partial Derivatives Calculus Index