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