There are rules we can follow to find many derivatives.

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".
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’ f )/g2 |
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 ddx ) | dydx = dydududx |
"The derivative of" is also written ddx
So ddxsin(x) and sin(x)’ are the same thing, just written differently
Examples
Power Rule
Multiplication by constant
Sum Rule
Difference Rule
It doesn't have to be x, we can differentiate with respect to, for example, v:
Sum, Difference, Constant Multiplication And Power Rules
Product Rule
Reciprocal Rule
Chain Rule
Another way of writing the Chain Rule is: dydx = dydududx
Let's do the previous example again using that formula:
Another couple of examples of the Chain Rule:
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