DERIVATIVES
Derivative means slope of the curve. It means that derivative at a certain point of a function is the slope of the function at this point, which is to the slope of the tangent.
m: The slope of the tangent to the graph y = f (x) at point P (a, f(a))
Rules of Derivatives:
u = g (x) v = h (x) w = m (x)
| RULE | FORMULA | EXAMPLE |
| Constant Function Rule | f (x) = k f1 (x) = 0 |
f (x) = 5 f1 = (x) = 0 |
| Linear Function Rule | f (x) = x f1 (x) = 1 |
f (x) = x f1 = (x) = 1 |
| The Power Rule | f (x) = xn f1 (x) = nxn-1
|
f (x) = x3 f1 (x) = 3.x2 |
| The Constant Multiple Rule | f (x) = k.g (x) f1 (x) = k.g1 (x) |
f (x) = 3.x4 f1 (x) = 12.x3 |
| The Sum Rule | f (x) = p (x) + q (x)
f1 (x) = f1 (x) + f1 (x) |
f (x) = 4.x3 + 5.x2
f1 (x) = 12.x2 + 10.x |
| The Difference Rule | f (x) = p (x) - q (x)
f1 (x) = f1 (x) - f1 (x) |
f (x) = 4.x3 - 5.x2
f1 (x) = 12.x2 - 10.x
|
| The Product Rule | f (x) = u. v f1 (x) = u1. v + u. v1 |
f (x) = (x2 - 2x).(x3 + 4) f1 (x) = (2x -2).(x3 + 4) + (x2 - 2x).(3x2) f1 (x) = 5x4 - 8x3 + 8x - 8 |
| The Extended Product Rule | f (x) = u. v. w f1 (x) = u1. v . w + u . v1. w + u . v . w1 |
f (x) = (x2 - 2x).(x4 + 6).(3x
+ 5) f1 (x) = (2x - 2).(x4 + 6).(3x + 5) + (x2 - 2x).(4x3).(3x + 5) + (x2 - 2x).(x4 + 6).(3) |
| The Power of a Function Rule | f (x) = un f1 (x) = n . un-1. u1 |
f (x) = (x2 - 3x + 4)5 f1 (x) = 5. (x2 - 3x + 4)4. (x -3) |
| The Quotient Rule |
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