Techniques of Differentiation Notes 3.3

I. Positive Integer Powers, Multiples, Sums, and Differences A.) Th: If f(x) is a constant, PF:

B.) Th: The Power Rule: If n is a positive integer and, PF:

Notice, n terms of x n-1.

C.) Th: The Constant Multiple Rule: If k is a real number and f is differentiable at x, then PF:

D.) Th: The Sum/Difference Rule: If f and g are differentiable functions of x at x, then their sum and their difference are differentiable at ever point where f and g are differentiable PF:

II. Examples A.) Find the following derivatives:

III. Derivative Notation for Functions of x A.) Given u as a function of x, the derivative of u is written as follows:

IV. Horizontal Tangent Lines A.) Occur when the slope of the tangent line equals zero. B.) Determine if and where the following function has any horizontal tangent lines

IV. (cont) Using Technology C.) Determine if and where the following function has any horizontal tangent lines

V. Product Rule A.) Th: PF:

B.) Use the product rule to find the derivatives of the following functions:

C.) Let y = uv be the product of the functions u and v. Find y’(2) if u(2) = 3, u’(2) = -4, v(2) = 1, and v’(2) = 2.

VI. Quotient Rule A.) Th: PF:

B.) Use the quotient rule to find the derivatives of the following functions:

VII. Negative Exponent Thm: A.) For any integer n ≠ 0, B.) Find the following derivative using both the quotient rule and the negative exponent theorem.

VII. Higher Order Derivatives Often we can find the derivative of a derivative. This will tell us valuable information about a function which we will investigate at a later date. Notation for higher-order derivatives is as follows: 1 st Deriv.2 nd Deriv.3 rd Deriv.n th Deriv.