If f x 0 for all x in an interval I then tan x sec2 x d 1 ln x, x 0.
dx dx 1 x2 dx x d d 1 Increasing/Decreasing 2. cos x sin x sin 1 x ln x, x 0 f x is concave up on the interval I. Common Derivatives d d d x 1 csc x csc x cot x ax a x ln a Increasing/Decreasing Concave Up/Concave Down dx dx dx Critical Points d d d x c is a critical point of f x provided either Concave Up/Concave Down sin x cos x cot x csc 2 x ex ex dx dx dx 1.ĥ If f x 0 for all x in an interval I then 1. g g2 2e 9y e 3x y 2x y y cos y y 11 y 2 x 3 y 9e 2 x 9 y cos y 2 x 3 y 9e 2 x 9y cos y y 11 2e 2 x 9y 3x2 y 2. Quotient Rule This is the Chain Rule 2x 9 y 2x 9 y 2 2 3 11 2e 2 x 9 y 3 x 2 y 2. f g x f g x g x e2 x 9y 2 9y 3x2 y2 2 x3 y y cos y y 11. x n x n 1 Power Rule After differentiating solve for y. f g f x g x d n differentiate as normal and every time you differentiate a y you tack on a y (from the chain rule).Ĥ 6. dx will use the product/quotient rule and Derivatives of y will use the chain rule. Remember y y x here, so products/quotients of x and y 1. Basic Properties and Formulas If f x and g x are differentiable functions (the derivative exists), c and n are any real numbers, Implicit Differentiation d Find y if e 2 x 9y x3 y 2 sin y 11 x. and is defined as f x and is defined as dx dx n equation of the tangent line at x a is time x then f a is the velocity of f x f x, the derivative of the f n x f n 1.ģ X, the derivative of given by y f a f a x a. If f x is the position of an object at f x f x 2. 2 d f n dn f line to y f x at x a and the 3. m f a is the slope of the tangent change of f x at x a. f a is the instantaneous rate of Higher Order Derivatives The Second Derivative is denoted as The nth Derivative is denoted as 1. dx 1 f x Interpretation of the Derivative If y f x then, 2. sec f ( x ) f ( x ) sec f ( x ) tan f ( x ).Ģ Df dy d df dy dx f x dx f x y f x Df x f a y x a Df a d f x dx dx dx dx x a dx x a d 8. notations for derivative evaluated at x a. tan f x f x sec 2 f x If y f x then all of the following are If y f x all of the following are equivalent dx dx d f x d equivalent notations for the derivative. cos f x f x sin f x h 0 h dx dx d f x f x d 2. f x h f x d n n 1 d If y f x then the derivative is defined to be f x lim. 1 Calculus Cheat Sheet Calculus Cheat Sheet Derivatives Chain Rule Variants Definition and Notation The chain rule applied to some specific functions.
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