Chapter 15 Differentiation Rules

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Basic Differentiation RulesYeah Man, It Rules

• We’ve seen how Calculus can be a little tricky.

• This is where Calculus will get easy.

• Learning the first half dozen or so rules is a snap (4th derivative).

• Some derivative rules are easy, but involve a little bit of work to remember them.

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

0d cdx

example: 2y 0y

The derivative of a constant is zero.

We saw that if , .2y x 2y x

This is part of a pattern.

1n nd x nxdx

examples:

4f x x

34f x x

8y x

78y x

power rule

2)( xxf32)(' xxf

32

)( xxf

31

32)('

xxf

xxh )( 1)(' xh

How would you differentiate a radical function?( )f x x

12( )f x x 1 121'( )

2f x x

121 1'( )

2 2f x x

x

How would you differentiate a radical function?

3 2)( xxf

32

)( xxf

31

32)('

xxf

33 321

32)('

xxxf

d ducu cdx dx

examples:

1n nd cx cnxdx

constant multiple rule:

5 4 47 7 5 35d x x xdx

See the coefficient does not influence the process of differentiation. You just ignoreit until the final step when you multiply the

initial exponent and the coefficient.

A few constant examples.• Differentiate

y = 4x3.

• Y’ = 4 • 3 x2. y’ = 12x2.

• Differentiate 45 3

1xy 3

2

125'

xy

3 2

5'12

yx

(Each term is treated separately)

d ducu cdx dx

constant multiple rule:

sum and difference rules:

d du dvu vdx dx dx

d du dvu vdx dx dx

4 12y x x 34 12y x

4 22 2y x x 34 4dy x x

dx

Find the gradient of the following at x = 2.• Differentiate

2 4y xx

2 14y x x

22 4 1dy x xdx

22 4dy x xdx

2

42dy xdx x

2

42( ) 5(

22)

dydx

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