Find the derivative Find the derivative at the following point. Warm-up Find the derivative Find the derivative at the following point.

Table of Contents 10. Section 3.2 The Derivative as a Function

The Derivative as a Function Essential Question – What rules of differentiation will make it easier to calculate derivatives?

Notation for Derivative If derivative exists, we say it is differentiable

Power Rule Power Rule Bring down the exponent and subtract one from the exponent

More notation means find the derivative of x4 when x = -2

2 more rules Constant multiple Sum and Difference

Differentiating a polynomial

Derivative of ex

Example Find the equation of the tangent line to the graph of f(x) = 3ex -5x2 at x=2

What information does the derivative at a point tell us? Tells us whether the tangent line has a positive or negative slope Tells us how steep the line is (the larger the derivative, the steeper the line) Tells us if there is a turning point (slope is 0)

Horizontal Tangents Does y = x4 – 2x2 + 2 have any horizontal tangents? First find the derivative, then set = 0 (because the slope of a horizontal line is 0)

Calculator example Find the points where horizontal tangents occur.

Graphing f’(x) from f(x) Find slope at each point Make a new graph using same x points and the slope as the y point If f is increasing, f ‘ will be positive (above the x axis) If f has a turning point, f ‘ will be 0 If f is decreasing, f ‘ will be negative (below the x axis)

Graph example Given the graph of f(x), which of A or B is the derivative?

Differentiability Differentiability implies continuity If f is differentiable at x = c, then f is continuous at x = c The opposite is not true A function can be continuous at x = c, but not differentiable

4 times a derivative fails to exist Corner Cusp

4 times a derivative fails to exist Vertical tangent Discontinuity

Local Linearity A function that is differentiable closely resembles its own tangent line when viewed very closely In other words, when zoomed in on a few times, a curve will look like a straight line.

Example

Assignment Pg. 139 #1-11 odd, 17-29 odd, 47-57 odd, 71-75 odd, 76-80 all, 83-87 odd