Derivative Rules
The Derivative tells us 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 | x^{2} | 2x |
Square Root | √x | (½)x^{-½} |
Exponential | e^{x} | e^{x} |
a^{x} | a^{x}(ln a) | |
Logarithms | ln(x) | 1/x |
log_{a}(x) | 1 / (x ln(a)) | |
Trigonometry (x is in radians) | sin(x) | cos(x) |
cos(x) | −sin(x) | |
tan(x) | sec^{2}(x) | |
Inverse Trigonometry | sin^{-1}(x) | 1/√(1−x^{2}) |
cos^{-1}(x) | −1/√(1−x^{2}) | |
tan^{-1}(x) | 1/(1+x^{2}) | |
Rules | Function |
Derivative |
Multiplication by constant | cf | cf’ |
Power Rule | x^{n} | nx^{n−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 )/g^{2} |
Reciprocal Rule | 1/f | −f’/f^{2} |
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 \frac{d}{dx} ) | \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx} |
"The derivative of" is also written \frac{d}{dx}
So \frac{d}{dx}sin(x) and sin(x)’ are the same thing, just written differently
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 x^{3} ?
The question is asking "what is the derivative of x^{3}?"
We can use the Power Rule, where n=3:
x^{n} = nx^{n−1}
x^{3} = 3x^{3−1} = 3x^{2}
Example: What is (1/x) ?
1/x is also x^{-1}
We can use the Power Rule, where n = −1:
x^{n} = nx^{n−1}
x^{−1} = −1x^{−1−1} = −x^{−2}
Multiplication by constant
Example: What is 5x^{3 }?
the derivative of cf = cf’
the derivative of 5f = 5f’
We know (from the Power Rule):
x^{3} = 3x^{3−1} = 3x^{2}
So:
5x^{3} = 5x^{3} = 5 × 3x^{2} = 15x^{2}
Sum Rule
Example: What is the derivative of x^{2}+x^{3 }?
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:
- x^{2} = 2x
- x^{3} = 3x^{2}
And so:
the derivative of x^{2} + x^{3} = 2x + 3x^{2}
Difference Rule
It doesn't have to be x, we can differentiate with respect to, for example, v:
Example: What is (v^{3}−v^{4}) ?
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:
- v^{3} = 3v^{2}
- v^{4} = 4v^{3}
And so:
the derivative of v^{3} − v^{4} = 3v^{2} − 4v^{3}
Sum, Difference, Constant Multiplication And Power Rules
Example: What is (5z^{2} + z^{3} − 7z^{4}) ?
Using the Power Rule:
- z^{2} = 2z
- z^{3} = 3z^{2}
- z^{4} = 4z^{3}
And so:
(5z^{2} + z^{3} − 7z^{4}) = 5 × 2z + 3z^{2} − 7 × 4z^{3} = 10z + 3z^{2} − 28z^{3}
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)
= cos^{2}(x) − sin^{2}(x)
Reciprocal Rule
Example: What is (1/x) ?
The Reciprocal Rule says:
the derivative of 1/f = −f’/f^{2}
With f(x)= x, we know that f’(x) = 1
So:
the derivative of 1/x = −1/x^{2}
Which is the same result we got above using the Power Rule.
Chain Rule
Example: What is \frac{d}{dx}sin(x^{2}) ?
sin(x^{2}) is made up of sin() and x^{2}:
- f(g) = sin(g)
- g(x) = x^{2}
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:
\frac{d}{dx}sin(x^{2}) = cos(g(x)) (2x)
= 2x cos(x^{2})
Another way of writing the Chain Rule is: \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}
Let's do the previous example again using that formula:
Example: What is \frac{d}{dx}sin(x^{2}) ?
\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}
Have u = x^{2}, so y = sin(u):
\frac{d}{dx} sin(x^{2}) = \frac{d}{du}sin(u)\frac{d}{dx}x^{2}
Differentiate each^{}:
\frac{d}{dx} sin(x^{2}) = cos(u) (2x)
Substitue back u = x^{2} and simplify:
\frac{d}{dx} sin(x^{2}) = 2x cos(x^{2})
Same result as before (thank goodness!)
Another couple of examples of the Chain Rule:
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/(g^{2})
- g'(x) = cos(x)
So:
(1/sin(x))’ = −1/(g(x))^{2} × cos(x)
= −cos(x)/sin^{2}(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 g^{3} and 5x-2:
- f(g) = g^{3}
- g(x) = 5x−2
The individual derivatives are:
- f'(g) = 3g^{2} (by the Power Rule)
- g'(x) = 5
So:
(5x−2)^{3} = 3g(x)^{2} × 5 = 15(5x−2)^{2}