2 Review from 3.2 4 places derivatives do not exist: Corner Cusp Vertical tangent (where derivative is undefined)Discontinuity (jump, hole, vertical asymptote, infinite oscillation)In other words, a function is differentiable everywhere in its domain if its graph is smooth and continuous.
3 3.2 Intermediate Value Theorem for Derivatives If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).
4 Derivatives of Constants Find the derivative of f(x) = 5.Derivative of a Constant:If f is the function with the constant value c, then,(the derivative of any constant is 0)
5 Power Rule If n is any real number and x ≠ 0, then What is the derivative of f(x) = x3?From class the other day, we know f’(x) = 3x2.If n is any real number and x ≠ 0, thenIn other words, to take the derivative of a term with a power, move the power down front and subtract 1 from the exponent.
6 Power Rule Example: Example: What is the derivative of
7 Power RuleExample:What is the derivative ofNow, use power rule
8 Constant Multiple Rule Find the derivative of f(x) = 3x2.Constant Multiple Rule:If u is a differentiable function of x and c is a constant, thenIn other words, take the derivative of the function and multiply it by the constant.
9 Sum/Difference Rule Find the derivative of f(x) = 3x2 + x If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points,In other words, if functions are separated by + or –, take the derivative of each term one at a time.
10 ExampleFind where horizontal tangent occurs for the function f(x) = 3x3 + 4x2 – 1.A horizontal tangent occurs when the slope (derivative) equals 0.
11 Example At what points do the horizontal tangents of f(x)=0.2x4 – 0.7x3 – 2x2 + 5x + 4 occur?Horizontal tangents occur when f’(x) = 0To find when this polynomial = 0, graph it and find the roots.Substituting these x-values back into the original equation gives us the points (-1.862, ), (0.948, 6.508), (3.539, )
12 Product Rule If u and v are two differentiable functions, then Also written as:In other words, the derivative of a product of two functions is “1st times the derivative of the 2nd plus the 2nd times the derivative of the 1st.”
14 Quotient RuleIf u and v are two differentiable functions and v ≠ 0, thenAlso written as:In other words, the derivative of a quotient of two functions is “low d-high minus high d-low all over low low.”