Derivatives Review: THINGS YOU NEED TO KNOW!!!

Basics: The derivative of any constant is always 0!!!!!

d d d f x  g x    f x    g x   all that this means is that derivatives distribute dx  dx   dx   over addition and subtraction

A numeric interpretation of the derivative is that it is the slope of the line tangent to a curve at a given value of x.

Power Rule: d du Trigonometric Rules: unn nu 1  d du dx dx cscu   csc u cot u  dx dx Logaritmic and Exponential Rules: d du tanuu sec2 d du   eeuu dx dx dx dx d du cotuu  csc2  d1 du   ln u  dx dx dx u dx d du cosuu   sin  dx dx Product Rule: d du d dv du secu  sec u tan u u v  u   v dx dx dx dx dx d du sinuu  cos dx dx Quotient Rule:

du dv vu du  dx dx  2 dx v v Inverse Trigonometric Rules: d1 du tan1 u  Chain Rule: dx1 u2 dx d f g x  f'' g x g x d1 1 du dx  sin u  dx1u2 dx

Less important, but it couldn’t hurt to know: d du auu aln a , where a is any constant that is not e. dx dx d1 du log u , for any logarithm – we just don’t see these that often. dxa uln a dx

Using Math 8  If you want to find a derivative at a specific value of x quickly, use Math 8 o Format: nDeriv(type in the function, X, value of X you are trying to find)  You can also graph a derivative this way. In the graph screen, type the following: o Y=(type in the function, X, X)

Definition of the Derivative: There are two limit definitions for the derivative: f x  f a f x h f x fa'   lim fx'   lim xa xa h0 h Both of these could be solved with L’Hospital’s rule, or by figuring out what fx  is, and taking its derivative. Just remember, when using L’Hospital’s Rule, the variable is indicated under the limit (either x or h). Any other letter is treated as a constant.

Interpretations of the Derivative:

 The first derivative: Tells you whether the original function is increasing (if f ‘ is positive) or decreasing (if f ‘ is negative). Generally the first derivative is velocity if the original function is position.

 The second derivative: Tells you if the original function is concave up (if f “ is positive) or concave down (if f “ is negative). Generally the second derivative is acceleration if the original function is position. The slope of a secant line: The slope of a line through two points on a curve approximates the derivative. There are three different ways of doing this:

x 4 6 8

fx  10 15 17

If you wanted to approximate f '6  , you could 17 10 a) use the slope between (4,10) and (8,17); f ' 6  1.75 84 15 10 b) use the slope between (4,10) and (6,15); f ' 6  2.5 64 17 15 c) use the slope between (6,15) and (8,17); f ' 6  1 86 Generally speaking, method a) yields the most accurate estimate.

Tangent Lines: The derivative is the slope of the tangent line, just plug in x to the derivative to find the slope, plug it into the original equation to find y, then plug all those in to the slope- intercept form of a line: y y11  m x  x  . To estimate a value for a function from its tangent line, first make a tangent line then plug in the x value to the tangent line. The y value that results is the approximate value for f.