Derivatives

What is a derivative?

• Mathematically, it is the slope of the tangent line at a given pt.

• Scientifically, it is the instantaneous velocity of a particle along a line at time, t.

• Or the instantaneous rate of change of a fnc. at a pt.

run

rise

x

y

dx

dy

Formal Definition of a Derivative:

0

limh

f a h f a

h

is called the derivative of f at a.

We write: 0

limh

f a h f af x

h

“The derivative of f with respect to x is …”

There are many ways to write the derivative of y f x

f x “f prime x” or “the derivative of f with respect to x”

y “y prime”

dy

dx“dee why dee ecks” or “the derivative of y with

respect to x”

df

dx“dee eff dee ecks” or “the derivative of f with

respect to x”

df x

dx“dee dee ecks uv eff uv ecks” or “the derivative

of f of x”( of of )d dx f x

dx does not mean d times x !

dy does not mean d times y !

dy

dx does not mean !dy dx

(except when it is convenient to think of it as division.)

df

dxdoes not mean !df dx

(except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.)

df x

dxdoes not mean times !

d

dx f x

A function is differentiable if it has a derivative everywhere in its domain. The limit must exist and it must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

p

y f x

y f x

The derivative is the slope of the original function.

The derivative is defined at the end points of a function on a closed interval.

2 3y x

2 2

0

3 3limh

x h xy

h

2 2 2

0

2limh

x xh h xy

h

2y x

0lim 2h

y x h

0

DIFFERENTIATION RULES:1. If f(x) = c, where c is a constant, then f’(x) = 02. If f(x) = c*g(x), then f’(x) = c*g’(x)3. If f(x) = xn, then f’(x) = nxn-1

4. SUM RULE: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x)

5. DIFFERENCE RULE:If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x)

6. PRODUCT RULE:If f(x) = g(x) * h(x), then f’(x) = g’(x)*h(x) +

h’(x)*g(x)7. QUOTIENT RULE:

f(x) = then f’(x) =

8. CHAIN RULE:If f(x) = g(h(x)), then f’(x) = g’(x)*h’(x)

,)(

)(

xh

xg2))((

)(*)(')(*)('

xh

xgxhxhxg

Derivatives to memorize:

• If f(x) = sin x, then f’(x) = cos x• If f(x) = cos x, then f’(x) = -sin x• If f(x) = tan x, then f’(x) = sec2x• If f(x) = cot x, then f’(x) = -csc2x• If f(x) = sec x, then f’(x) =

secxtanx• If f(x) = csc x, then f’(x) = -

cscxcotx• If f(x) = ex, then f’(x) = ex

• If f(x) = ln x, then f’(x) = 1/x• If f(x) = ax, then f’(x) = (ln a) * ax

Examples: Find f’(x) if 1. f(x) = 52. f(x) = x2 – 53. f(x) = 6x3+5x2+9x+34. f(x) = (3x+4)(2x2-3x+5)5. f(x) =

6. f(x) =

7. f(x) = (3x2+5x-2)8

8. f(x)=

32

53

x

x

2

4

x

135 2 xx