DO NOW: Find the equation of the line tangent to the curve f(x) = 3x 2 + 4x at x = -2.

DO NOW: Find the equation of the line tangent to

the curve f(x) = 3x2 + 4x at x = -2

3.1 – DERIVATIVE OF A FUNCTION

Remember… The slope of the line tangent to a curve

with equation y = f(x) at x = a:

Similarly, the velocity of an object with position function s = f(t) at t = a:

hafhafm

h

)()(lim0

hafhafav

h

)()()( lim0

The DerivativeDEFINITION The derivative of a function f at a

number x=a, denoted by f’(a), is

If this limit exists.

Also written:

hafhafaf

h

)()()(' lim0

axafxfaf

ax

)()()(' lim

Example 1 Find the derivative of the function

f(x) = x2 – 8x + 9 at the number a.

Interpretation as Slope of Tangent The tangent line to y = f(x) at (a, f(a)) is

the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a.

The next two slides illustrate this interpretation of the derivative:

Slope of Tangent (2 interpretations)

Tangent Line Through a Point Using the point-slope form of the

equation of a line, we can write an equation of the line tangent to the curve y = f(x) at point (a, f(a)):

y – f(a) = f’(a)(x – a)

Example 2 Find an equation of the line tangent to

f(x) = x2 – 8x + 9 at the point (3,-6)

Interpretation as Rate of Change

The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.

Note that when the derivative is… Large, the y-values change

rapidly; Small, the y-values change

slowly.

Velocity and Speed Position Function: s = f(t)

Position along straight line

Velocity Function: f’(a) Velocity of the function at t = a

Speed: |v(t)|

Example 3 The position of a particle is given by the

equation of motion s = f(t) = 1/(1 + t), where t is measured in seconds and s in meters.

Find the velocity and speed after 2 seconds.

Derivative So far we have considered the derivative

of a function f at a fixed number a:

Now we change our point of view and let the number a vary:

hafhafaf

h

)()()(' lim0

hxfhxfxf

h

)()()(' lim0

Notation

y’“y prime” Nice and

brief, but does not name the independent variable

“dy dx” or “the derivative of y with respect to x”

Names both variables and uses d for derivative.

“df dx” of “the derivative of f with respect to x”

Emphasizes the function’s name

“d dx of f at x” or “the derivative of f at x;

Emphasizes the idea that differentiation is an operation performed on f.

dxdy

dxdf

)(xfdxd

Example 4 At right is the graph of a function f. Use

this graph to sketch the graph of the derivative f’(x).

Example 4 (solution)

Example 5 For the function f(x) = x3 – x

Find a formula for f’(x) Compare the graphs of f and f’

Example 5 (solution)

One-Sided Derivatives A function y = f(x) is differentiable on a

closed interval [a,b] if it has a derivative at every interior point of the interval, and if the limits[the right-hand derivative at a]

[the left-hand derivative at a]

Exist at the endpoints.

hafhaf

h

)()(lim0

hbfhbf

h

)()(lim0

Example 6 Show that the following function has left-

hand and right-hand derivatives at x=0, but no derivative there.

,2,2

xx

y 00

xx

Example 6 (solution)

Practice: Pg. 205 #1-19odd