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Differentiation Formulas. Let's start with the simplest of all functions, the constant function f(x) = c. The graph of this function is the horizontal.

Differentiation Formulas d dx k = 0 (1) d dx [f(x)±g(x)] = f0(x)±g0(x) (2) d dx [k ·f(x)] = k ·f0(x) (3) d dx [f(x)g(x)] = f(x)g0(x)+g(x)f0(x) (4) d dx f(x) g(x ...

Basic Derivative formulae. (xn) = nxn−1. (ax) = ax ln a ... If y = y(x) is given implicitly, find derivative to the entire equation with respect to x.

• Trig Derivatives: – f(x)=sin(x)thenf0(x)=cos(x) – f(x)=cos(x)thenf0(x)=−sin(x) – f(x)=tan(x)thenf0(x)=sec2(x) – f(x)=sec(x)thenf0(x)=sec(x)tan(x) – f(x)=cot(x)thenf0(x)=−csc2(x) – f(x)=csc(x)thenf0(x)=−csc(x)cot(x) • Exponential Derivatives – f(x)=ax thenf0(x)=ln(a)ax – f(x)=ex thenf0(x)=ex – f(x)=ag(x) thenf0(x)=ln(a)ag(x)g0(x)

ListofDerivativeRules Belowisalistofallthederivativeruleswewentoverinclass. • Constant Rule: f(x)=cthenf0(x)=0 • Constant Multiple Rule: g(x)=c·f(x)theng0(x)=c ...

Limits Derivatives Math Formulas Higher-order Created Date: 1/31/2010 3:27:33 AM ...

Differentiation Formulas d dx k = 0 (1) d dx [f(x)±g(x)] = f0(x)±g0(x) (2) d dx [k ·f(x)] = k ·f0(x) (3) d dx [f(x)g(x)] = f(x)g0(x)+g(x)f0(x) (4) d dx f(x) g(x) = g(x)f0(x)−f(x)g0(x) [g(x)]2 (5) d dx f(g(x)) = f0(g(x))·g0(x) (6) d dx xn = nxn−1 (7) d dx sinx = cosx (8) d dx cosx = −sinx (9) d dx tanx = sec2 x (10) d dx cotx = −csc2 x (11) d dx secx = secxtanx (12) d dx

2. Common Derivatives Basic Properties and Formulas ( )cf cf x′ = ′( ) ( )f g f x g x± = +′ ′ ′( ) ( ) Product rule ( )f g f g f g⋅ = ⋅ + ⋅′ ′ ′ Quotient rule 2 f f g f g g g ′ ′ ′⋅ − ⋅ = Power rule d (x nxn n) 1 dx = − Chain rule ( )( )( ) ( )( ) ( ) d f g x f g x g x dx = ′ ′ Common Derivatives ( ) 0 d c dx = ( ) 1 d x dx = ( )sin cos d x x dx = ( )cos sin d x x dx

Limits and Derivatives Formulas. 1. Limits. Properties if lim ( ). x a. f x l. →. = and lim ( ). x a. g x m. →. = , then. [. ] lim ( ). ( ). x a. f x g x.

Basic Differentiation Formulas. In the table below, and represent differentiable functions of ? œ 0РBС. @ œ 1РBС. B. Derivative of a constant.

Derivatives Definition and Notation If yfx then the derivative is defined to be 0 lim h fx h fx fx h . If yfx then all of the following are equivalent notations for the derivative. fx y fx Dfx df dy d dx dx dx If yfx all of the following are equivalent notations for derivative evaluated at x a.

Differentiation Formulas The following table provides the differentiation formulas for common functions. The first six rows correspond to general rules (such as the addition rule or the product rule) whereas the remaining rows contain the formulas for specific functions.

Higher Order Derivatives The Second Derivative is denoted as 2 2 2 df fx f x dx and is defined as fx fx , i.e. the derivative of the first derivative, fx . The nth Derivative is denoted as n n n df fx dx and is defined as fx f x nn 1 , i.e. the derivative of

Differentiation Formulas d dx k = 0. (1) d dx. [f(x) ± g(x)] = f (x) ± g (x). (2) d dx. [k · f(x)] = k · f (x). (3) d dx. [f(x)g(x)] = f(x)g (x) + g(x)f (x) ...

1. Basic Derivative formulae (xn)0 = nxn−1 (ax)0 = ax lna (ex)0 = ex (log a x) 0 = 1 xlna (lnx)0 = 1 x (sinx)0 = cosx (cosx)0 = −sinx (tanx)0 = sec2 x (cotx)0 = −csc2 x (secx)0 = secxtanx (cscx)0 = −cscxcotx (sin−1 x)0 = 1 √ 1−x2 (cos−1 x)0 = −1 √ 1−x2 (tan−1 x)0 = 1 1+x2 (cot−1 x)0 = −1 1+x2 (sec−1 x)0 = 1 x √ x2 −1 (csc−1 x)0 = −1 x √ x2 −1 2 ...

Apart from these formulas, PDF also covered the derivatives of trigonometric functions and inverse trigonometric functions as well as rules of differentiation.

Basic Differentiation Formulas In the table below, and represent differentiable functions of ?œ0ÐBÑ @œ1ÐBÑ B Derivative of a constant.-.B œ! Derivative of constan t ( ) We could also write , and could use..?.B .B-? œ- Ð Ð-0Ñœ-0ww the “prime notion” in …

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Use double angle formula for sine and/or half angle formulas to reduce the integral into a form that can be integrated. tan sec n m x x dx. ∫.

Differentiation Formulas List. 1. Basic Differentiation Formulas. Page 2. 2. Differentiation Formulas For Trigonometric Functions. Page 3 ...

1. Basic Derivative formulae (xn)0 = nxn−1 (ax)0 = ax lna (ex)0 = ex (log a x) 0 = 1 xlna (lnx)0 = 1 x (sinx)0 = cosx (cosx)0 = −sinx (tanx)0 = sec2 x (cotx)0 = −csc2 x (secx)0 = secxtanx (cscx)0 = −cscxcotx (sin−1 x)0 = 1 √ 1−x2 (cos−1 x)0 = −1 √ 1−x2 (tan−1 x)0 = 1 1+x2 (cot−1 x)0 = −1 1+x2 (sec−1 x)0 = 1 x √ x2 −1 (csc−1 x)0 = −1 x √ x2 −1 2. Differentiation Rules

List of Derivative Rules. Below is a list of all the derivative rules we went over in class. • Constant Rule: f(x) = c then f (x)=0.