3-2 The Derivative Thurs Sept 24 Find the slope of the tangent line to y = f(x) at x = a 1)x^2 -4, a = 2 2)2x^3, a = 0.

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Presentation transcript:

3-2 The Derivative Thurs Sept 24 Find the slope of the tangent line to y = f(x) at x = a 1)x^2 -4, a = 2 2)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) 10 X^11 X^22x X^33x^2 X^44x^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 3.1 Find the derivatives of

Ex 3.2 Find 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