Here are useful rules to aid you occupational out the derivatives of countless functions (with instances below). Note: the little mark ’ method derivative of, and f and also g space functions.
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| 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) | |
| 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 that Functions") | f º g | (f’ º g) × g’ |
| Chain dominance (using ’ ) | f(g(x)) | f’(g(x))g’(x) |
| Chain preeminence (using ddx ) | dydx = dydududx |
"The derivative of" is additionally written ddx
So ddxsin(x) and sin(x)’ both average "The derivative the sin(x)"
Example: what is the derivative that sin(x) ?
From the table over it is detailed as gift cos(x)
It deserve to be composed as:
ddxsin(x) = cos(x)
Or:
sin(x)’ = cos(x)
Example: What is ddxx3 ?
The inquiry is asking "what is the derivative of x3 ?"
We have the right to use the strength Rule, whereby n=3:
ddxxn = nxn−1
ddxx3 = 3x3−1 = 3x2
(In various other words the derivative of x3 is 3x2)
So that is just this:
It can also be offered in situations like this:
Example: What is ddx(1/x) ?
1/x is additionally x-1
We have the right to use the strength Rule, wherein n = −1:
ddxxn = nxn−1
ddxx-1 = −1x-1−1
= −x-2
= −1x2
So we just did this:
Multiplication by constant
Example: What is ddx5x3 ?
the derivative that cf = cf’
the derivative of 5f = 5f’
We know (from the power Rule):
ddxx3 = 3x3−1 = 3x2
So:
ddx5x3 = 5ddxx3 = 5 × 3x2 = 15x2
See more: What Is The Prime Factorization Of 13 As A Product Of Prime Numbers?
Example: What is the derivative that x2+x3 ?
The Sum dominance says:
the derivative of f + g = f’ + g’
So we deserve to work out each derivative separately and then add them.
Using the power Rule:
ddxx2 = 2xddxx3 = 3x2And so:
the derivative that x2 + x3 = 2x + 3x2
Difference Rule
What we identify with respect come doesn"t have to be x, it could be anything. In this case v: