3-2 The DerivativeThurs Sept 24

Find the slope of the tangent line to y = f(x) at x = a

1) x^2 -4, a = 22) 2x^3, a = 0

Alternative derivative notations

• There are several ways to denote the derivative. We already know f’(x)

• The following notations are all equivalent:

• These notations indicate “the derivative of y in terms of x”

Differentiability and Continuity

• If f is differentiable at x = c (the derivative is defined at c) then f is also continuous at c

When a function is not differentiable at a point

• When a function is not differentiable at a point x = a, the one sided limits will not be equal. There are several cases:

• A jump discontinuity (piecewise function)• Vertical asymptote• Cusp (piecewise function)• Vertical tangent line

Derivative Info

• The derivative can tell us when a function is increasing (+), decreasing (-), or horizontal (0)

• This makes finding the vertex of a function easier

Benefits of the Derivative

• The derivative also gives us a good view of the behavior of the original function f(x)

• Slope• Velocity• Rates of change

Computations of Derivatives

• Thm- For any constant c,

• Note, when y = c, the slope of that line is always horizontal. Therefore, its derivative must equal 0

• Thm- Let f(x) = x, then

• Proof:

• Note: This means that the derivative of any linear function is equal to the coefficient

Power Rule• Let’s take a look at the different powers of x. Can you

see the pattern in the table?

F(x) F’(x)

1 0

X^1 1

X^2 2x

X^3 3x^2

X^4 4x^3

Power Rule cont’d

Power Rule - For any real number n,

Note: The power rule works for negative exponents, as well as fraction exponents.

Ex 3.1Find the derivatives of

Ex 3.2Find the derivatives of

Derivative of e^x

• The derivative of f(x) = e^x is

General Derivative Rules

• Thm- If f(x) and g(x) are differentiable at x and c is any constant, then

• 1)

• 2)

• 3)

General Deriv. Rules• Remember, to rewrite any expressions so

they have exponents! And split the expression into separate terms!

• You try: Find the derivative of each:• 1)

• 2)

• 3)

Closure

• Hand in: Find the derivative of:• 1)

• 2)

• HW: p.139 #7, 13, 17, 25, 37, 43, 49