Example: Later, though, we will meet functions, such as y = x 2 sinx, for which the product rule is the only possible method.

example:

The Quotient Rule can be used to extend the Power Rule to the case where the exponent is a negative integer. Example: ▪ If n = 0, then x 0 = 1, which we know has a derivative of 0. Thus, the Power Rule holds for any integer n. What if the exponent is a fraction? ▪ In fact, it can be shown by using Chain Rule (obviously proof later) that it also holds for any real number n.

The derivatives of the remaining trigonometric functions — csc, sec, and cot — can also be found easily using the Quotient Rule. All together:

Example: Differentiate For what values of x does the graph of f have a horizontal tangent?

Since sec x is never 0, we see that f’(x) = 0 when tan x = 1. This occurs when x = nπ + π/4, where n is an integer

Example: Find Example: Calculate:

Intermediate Value Theorem for Continuous Functions As example on our graph:

You can find it in this form too: Intermediate Value Theorem for Derivatives

Higher Order Derivatives: Just as one can obtain a velocity function by differentiating a position function, one can obtain an acceleration function by differentiating a velocity function. Alternatively, one can think about obtaining an acceleration function by differentiating a position function twice.

Higher Order Derivatives:

From a general definition of the derivative: FORM A Example: From the definition of the derivative at point c: FORM B