- Slides: 18
3. 3 Rules for Differentiation AKA “Shortcuts”
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. 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) = 3 x 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) = 3 x 2. Constant Multiple Rule: If u is a differentiable function of x and c is a constant, then 0 In other words, take the derivative of the function and multiply it by the constant.
Sum/Difference Rule • Find the derivative of f(x) = 3 x 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) = 3 x 3 + 4 x 2 – 1. A horizontal tangent occurs when the slope (derivative) equals 0.
Example • At what points do the horizontal tangents of f(x)=0. 2 x 4 – 0. 7 x 3 – 2 x 2 + 5 x + 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, -5. 321), (0. 948, 6. 508), (3. 539, -3. 008)
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. ”
Quotient Rule • Example: Find the derivative of
Higher-Order Derivatives • f’ f(n)isiscalledthe thefirst nth derivative of f • f'' is called the second derivative of f • f''' is called the third derivative of f
Higher-Order Derivatives • Example Find the first four derivatives of
Friday Classwork: • Section 3. 3 ▫ (#1 -9 odd, 15 -23 odd, 25, 27, 29, 33 -37 odd, 46)