1. Derivatives

2. What is a derivative? Mathematically, it is the slope of the tangent line at a given pt. Scientifically, it is the instantaneous velocity of a particle along a line at time, t. Or the instantaneous rate of change of a fnc. at a pt.

3. Formal Definition of a Derivative: is called the derivative of f at a. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative of

4. “f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” “dee dee ecks uv eff uv ecks”or “the derivative of f of x”

5. dx does not mean d times x ! dy does not mean d times y !

6. does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

7. (except when it is convenient to treat it that way.) does not mean times !

8. A function is differentiable if it has a derivative everywhere in its domain. The limit must exist and it must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. 

9. The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

11. DIFFERENTIATION RULES: 1.If f(x) = c, where c is a constant, then f’(x) = 0 2.If f(x) = c*g(x), then f’(x) = c*g’(x) 3.If f(x) = x n, then f’(x) = nx n-1 4.SUM RULE: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) 5.DIFFERENCE RULE: If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x) 6.PRODUCT RULE: If f(x) = g(x) * h(x), then f’(x) = g’(x)*h(x) + h’(x)*g(x) 7.QUOTIENT RULE: f(x) = then f’(x) = 8.CHAIN RULE: If f(x) = g(h(x)), then f’(x) = g’(x)*h’(x)

12. Derivatives to memorize: If f(x) = sin x, then f’(x) = cos x If f(x) = cos x, then f’(x) = -sin x If f(x) = tan x, then f’(x) = sec 2 x If f(x) = cot x, then f’(x) = -csc 2 x If f(x) = sec x, then f’(x) = secxtanx If f(x) = csc x, then f’(x) = -cscxcotx If f(x) = e x, then f’(x) = e x If f(x) = ln x, then f’(x) = 1/x If f(x) = a x, then f’(x) = (ln a) * a x

13. Examples: Find f’(x) if 1.f(x) = 5 2.f(x) = x 2 – 5 3.f(x) = 6x 3 +5x 2 +9x+3 4.f(x) = (3x+4)(2x 2 -3x+5) 5.f(x) = 6.f(x) = 7.f(x) = (3x 2 +5x-2) 8 8.f(x)=