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 tan1 u Chain Rule: dx1 u2 dx d f g x f'' g x g x d1 1 du dx sin u dx1u2 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 h0 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. .