The slope of a constant value (like 3) is always 0The slope of a line like 2x is 2, or 3x is 3 etcand 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.




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Common FunctionsFunctionDerivativeRulesFunctionDerivative
Constantc0
Linex1
axa
Squarex22x
Square Root√x(½)x-½
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)
Multiplication by constantcfcf’
Power Rulexnnxn−1
Sum Rulef + gf’ + g’
Difference Rulef - gf’ − g’
Product Rulefgf g’ + f’ g
Quotient Rulef/gf’ g − g’ fg2
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)"


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)


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:

*
3x^2" style="width:66px; height:107px; min-width:66px;">"multiply by powerthen 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:

*
-x^-2" style="width:73px; height:107px; min-width:73px;">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




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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 = 2xddxx3 = 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: