Calculus Cheat Sheet Derivatives, Cheat Sheet of Calculus

Download Calculus Cheat Sheet Derivatives and more Cheat Sheet Calculus in PDF only on Docsity! Calculus Cheat Sheet Derivatives Definition and Notation If ( )y f x= then the derivative is defined to be ( ) ( ) ( ) 0 lim h f x h f x f x h→ + − ′ = . If ( )y f x= then all of the following are equivalent notations for the derivative. ( ) ( )( ) ( )df dy df x y f x Df x dx dx dx ′ ′= = = = = If ( )y f x= all of the following are equivalent notations for derivative evaluated at x a= . ( ) ( )x a x a x a df dyf a y Df a dx dx= = = ′ ′= = = = Interpretation of the Derivative If ( )y f x= then, 1. ( )m f a′= is the slope of the tangent line to ( )y f x= at x a= and the equation of the tangent line at x a= is given by ( ) ( )( )y f a f a x a′= + − . 2. ( )f a′ is the instantaneous rate of change of ( )f x at x a= . 3. If ( )f x is the position of an object at time x then ( )f a′ is the velocity of the object at x a= . Basic Properties and Formulas If ( )f x and ( )g x are differentiable functions (the derivative exists), c and n are any real numbers, 1. ( ) ( )c f c f x′ ′= 2. ( ) ( ) ( )f g f x g x′ ′ ′± = ± 3. ( )f g f g f g′ ′ ′= + – Product Rule 4. 2 f f g f g g g ′ ′ ′  − =    – Quotient Rule 5. ( ) 0d c dx = 6. ( ) 1n nd x n xdx −= – Power Rule 7. ( )( )( ) ( )( ) ( )d f g x f g x g xdx ′ ′= This is the Chain Rule Common Derivatives ( ) 1d x dx = ( )sin cosd x x dx = ( )cos sind x x dx = − ( ) 2tan secd x x dx = ( )sec sec tand x x x dx = ( )csc csc cotd x x x dx = − ( ) 2cot cscd x x dx = − ( )1 2 1sin 1 d x dx x − = − ( )1 2 1cos 1 d x dx x − = − − ( )1 21tan 1 d x dx x − = + ( ) ( )lnx xd a a adx = ( )x xddx =e e ( )( ) 1ln , 0d x x dx x = > ( ) 1ln , 0d x x dx x = ≠ ( )( ) 1log , 0 lna d x x dx x a = > Calculus Cheat Sheet Chain Rule Variants The chain rule applied to some specific functions. 1. ( )( ) ( ) ( )1n nd f x n f x f xdx − ′=       2. ( )( ) ( ) ( )f x f xd f xdx ′=e e 3. ( )( ) ( )( )ln f xd f x dx f x ′ =   4. ( )( ) ( ) ( )sin cosd f x f x f xdx ′=       5. ( )( ) ( ) ( )cos sind f x f x f xdx ′= −       6. ( )( ) ( ) ( )2tan secd f x f x f xdx ′=       7. [ ]( ) [ ] [ ]( ) ( ) ( ) ( )sec sec tanf x f x f x f xd dx ′= 8. ( )( ) ( ) ( ) 1 2tan 1 f xd f x dx f x − ′=   +    Higher Order Derivatives The Second Derivative is denoted as ( ) ( ) ( ) 2 2 2 d ff x f x dx ′′ = = and is defined as ( ) ( )( )f x f x ′′′ ′= , i.e. the derivative of the first derivative, ( )f x′ . The nth Derivative is denoted as ( ) ( ) n n n d ff x dx = and is defined as ( ) ( ) ( ) ( )( )1n nf x f x− ′= , i.e. the derivative of the (n-1)st derivative, ( ) ( )1nf x− . Implicit Differentiation Find y′ if ( )2 9 3 2 sin 11x y x y y x− + = +e . Remember ( )y y x= here, so products/quotients of x and y will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y′ (from the chain rule). After differentiating solve for y′ . ( ) ( ) ( ) ( )( ) ( ) 2 9 2 2 3 2 9 2 2 2 9 2 9 2 2 3 3 2 9 3 2 9 2 9 2 2 2 9 3 2 cos 11 11 2 32 9 3 2 cos 11 2 9 cos 2 9 cos 11 2 3 x y x y x y x y x y x y x y y x y x y y y y x yy x y x y y y y y x y y x y y y x y − − − − − − − ′ ′ ′− + + = + − −′ ′ ′ ′− + + = + ⇒ = − − ′− − = − − e ee e e e e Increasing/Decreasing – Concave Up/Concave Down Critical Points x c= is a critical point of ( )f x provided either 1. ( ) 0f c′ = or 2. ( )f c′ doesn’t exist. Increasing/Decreasing 1. If ( ) 0f x′ > for all x in an interval I then ( )f x is increasing on the interval I. 2. If ( ) 0f x′ < for all x in an interval I then ( )f x is decreasing on the interval I. 3. If ( ) 0f x′ = for all x in an interval I then ( )f x is constant on the interval I. Concave Up/Concave Down 1. If ( ) 0f x′′ > for all x in an interval I then ( )f x is concave up on the interval I. 2. If ( ) 0f x′′ < for all x in an interval I then ( )f x is concave down on the interval I. Inflection Points x c= is a inflection point of ( )f x if the concavity changes at x c= .