4. Product and Quotient Rules

Product Rule Multiplication and addition are both commutative so the order does not matter

Example 1 Find the derivative using the product rule, then verify using the power rule and graphically using NDERIV.

Example 2 Find the derivative of each of the following

Quotient Rule Subtraction is not commutative so this one is more important to get straight If you call the numerator “HI” and the denominator “LO”, and d is derivative, this becomes

Example 3 Find the derivative using the quotient rule, then verify graphically using NDERIV.

Example 4 Rewriting is still key. Find the derivative of each of the following using the most efficient method

Example 5 Find the derivative of using the quotient rule.

Derivatives of other trig functions

Example 6 Find the derivatives of each of the following

Example 7 Find the derivatives of each of the following, then show they are equivalent.

Higher order derivatives If you take the derivative twice it is called the second derivative. You can continue to take derivatives as long as you like.

Example 8 Find the first 5 derivatives of

What information does the derivative at a point tell us? Tells us whether the tangent line has a positive or negative slope Tells us how steep the line is (the larger the derivative, the steeper the line) Tells us if there is a turning point (slope is 0)

Example 9 The graph below is the graph of f. On the same coordinate grid, sketch and label the possible graphs of f’, f’’, f’’’.