Original Rule |
Generalized Rule (Chain Rule) |
Comments |
d
dx |
f(x) = g(x) |
|
d
dx |
f(u) = g(u) |
du
dx |
|
General form of Chain Rule |
d
dx |
xn = nx n-1 |
|
d
dx |
un = nun-1 |
du
dx |
|
Generalized Power Rule |
d
dx |
4x-1/2 = -2x-3/2 |
|
d
dx |
4u-1/2 = -2u-3/2 |
du
dx |
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An example of the above rule |
d
dx |
sin x = cos x |
|
d
dx |
sin u = cos u |
du
dx |
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Take me to text on trig functions! |
d
dx |
ln x |
= |
1
x |
|
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The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity. |
d
dx |
logb(x) |
= |
1
x ln(b) |
|
d
dx |
logb(u)
| = |
1
u ln(b) |
du
dx |
|
|
d
dx |
ex |
= |
ex |
|
d
dx |
eu |
= |
eu |
du
dx |
|
The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity. |
d
dx |
bx |
= |
bx ln(b) |
|
d
dx |
bu |
= |
bu ln(b) |
du
dx |
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