Lesson 3.3: Rules for Differentiability AP Calculus Mrs. Mongold

If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero.

If we find derivatives with the difference quotient: We observe a pattern: …

We observe a pattern: … examples: power rule

constant multiple rule: examples: When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

constant multiple rule: sum and difference rules: (Each term is treated separately)

Example: Find the horizontal tangents of: Horizontal tangents occur when slope = zero. Plugging the x values into the original equation, we get: (The function is even, so we only get two horizontal tangents.)

First derivative (slope) is zero at:

product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:

quotient rule: or

Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative. We will learn later what these higher order derivatives are used for. is the fourth derivative. p

You couldn’t wait for “shortcuts” Find the derivative of h(x)=(3x-2x2)(5+4x)

Find the derivative of f(x)=(3x3+4x)(x-5)(x+1)

 

 

Homework Page 120-121/1-12, 14-34 Even