Log/Exponent Properties: ln(1) = 0 ln(e) = 1 ln(a n ) = n*ln(a) ln(ab) = ln(a) + ln(b) b a b a ln ln ln − = Exponent Properties: e a * e b = e a+b (e a ) b = e ab e 0 = 1 Log Differentiation steps: 1) Take ln of both sides 2) Expand right side. 3) Find derivative 4) Solve for dy/dx Evaluate derivative of inverse: (find !! ! () 1.Set f(x) = a and solve for x (guess and check) 2. Find f ‘(x) 3. Plug in x value from step #1 into f ‘(x). 4. Flip value. Log Derivatives: Exponential Derivatives d dx ln | u | = u ' u d dx e u = e u ∗ u ' u u a u dx d a ' * ln 1 log = d dx a u = ln a ∗ a u * u ' Trig Derivatives: d dx sin u = cos u * u ' d dx tan u = sec 2 u * u ' d dx sec u = sec u tan u * u ' d dx cos u = − sin u * u ' d dx cot u = − csc 2 u * u ' d dx csc u = − csc u cot u * Inverse Trig Derivatives: d dx arcsin u = u ' 1 − u 2 d dx arctan u = u ' 1 + u 2 d dx arc sec u = u ' u u 2 − 1 d dx arccos u = − u ' 1 − u 2 d dx arc cot u = − u ' 1 + u 2 d dx arc csc u = − u ' u u 2 − 1 Integral Formulas: Power Rule: u n du = u n+ 1 n + 1 + C ∫ Log Rule: 1 u du = ln | u | +C ∫ Exponential Rule: (Base e) ∫ du e u = e u +C Exponential Rule (base other than e) a u du = a u ln a + C ∫ *Note: lna is a constant* Trig Integrals: sin udu = − cos u + C ∫ cos udu = sin u + C ∫ sec 2 udu = tan u + C ∫ sec u tan udu = sec u + C ∫ csc 2 udu = − cot u + C ∫ csc u cot udu = − csc u + C ∫ C + | cosu | -ln tan = ∫ udu C u udu + = ∫ sin ln cot ∫ udu sec = ln|sec u + tan u| + C ∫ udu csc = ln|csc u + cot u| + C Inverse Trig Integrals: ∫ + = − C a u u a du arcsin 2 2 ∫ + = + C a u a u a du arctan 1 2 2 ∫ + = − C a u a a u u du | | sec arc 1 2 2 !"# ! ! = and log ! ! = log ! = ln ln Interest Formulas = !1 + ! (!") A = Pe rt