3.3 Rules for Differentiation AKA “Shortcuts”

Review from 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.

3.2 Using the Calculator The calculator will find approximations for numerical derivatives. ▫Ex.: Find the slope of the tangent line of f(x) = x 3 at x = 5.

3.2 Differentiability Theorem: ▫If f has a derivative at x = a, then f is continuous at x = a. ▫If f is differentiable everywhere, it is also continuous everywhere.

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).

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)

Power Rule What is the derivative of f(x) = x 3 ? From class the other day, we know f’(x) = 3x 2. If n is any real number and x ≠ 0, then In other words, to take the derivative of a term with a power, move the power down front and subtract 1 from the exponent.

Power Rule Example: ▫What is the derivative of Example : – What is the derivative of

Power Rule Example: ▫What is the derivative of Now, use power rule

Constant Multiple Rule Find the derivative of f(x) = 3x 2. 0 Constant Multiple Rule: If u is a differentiable function of x and c is a constant, then In other words, take the derivative of the function and multiply it by the constant.

Sum/Difference Rule Find the derivative of f(x) = 3x 2 + x Sum/Difference Rule: 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.

Example Find where horizontal tangent occurs for the function f(x) = 3x 3 + 4x 2 – 1. A horizontal tangent occurs when the slope (derivative) equals 0.

Example At what points do the horizontal tangents of f(x)=0.2x 4 – 0.7x 3 – 2x 2 + 5x + 4 occur? Horizontal tangents occur when f’(x) = 0 To 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, )

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 “1 st times the derivative of the 2 nd plus the 2 nd times the derivative of the 1 st.”

Product Rule Example: Find the derivative of

Quotient Rule If u and v are two differentiable functions and v ≠ 0, then Also 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.”

Quotient Rule Example: Find the derivative of

Higher-Order Derivatives f’ is called the first derivative of f f'' is called the second derivative of f f''' is called the third derivative of f f (n) is called the n th derivative of f

Higher-Order Derivatives Example Find the first four derivatives of