Section 3.7“Derivatives of logarithmic functions”1Rules of exponentials and logarithms1. ab+c= abac2. ab−c=abac3. (ab)c= abc4. a−1=1a5. a0= 1.alogab= b1. loga(bc) = loga(b) + loga(c)2. logabc= loga(b) − loga(c)3. loga(bc) = c logab4. loga1b= − logab5. loga1 = 0.2Derivatives of logarithmic functionsTheorem. 1.ddxln x =1x.2. If a > 0,ddxlogax =1(ln a)x.3Derivatives of exponential functionsTheorem. If a is any positive number,ddxax= (ln a)ax.4Tips for Logarithmic Differentiation1. Start with y = f(x).2. Take the natural logarithm of both sides.3. Use properties of logarithms to simplify the right-hand side.4. Take the derivative. On the left you will haveddxln y =1ydydx.5. Multiply both sides by y and substitute y = f(x).5Theorem.limh→0ln(1 + h)1/h= 1.Proof. Let f(x) = ln x. We know f0(1) = 0. This means1 = limh→0f(1 + h) − f(1)h= limh→0ln(1 + h) − ln 1h= limh→0ln(1 + h)1/h.6Theorem.limn→∞1 +1nn= e.Proof. First, exponentiate the last theorem:elimh→0ln(1+h)1/h= e1limh→0eln(1+h)1/h= e1limh→0(1 + h)1/h= e.Now, if n is any positive number, let h =1n. Then as n → ∞, h → 0,and solimn→∞1 +1nn= limh→0(1 + h)1/h= e.7Questions81. [Q] True or False.ddxln(π) =1π.92. [Q] Your calculus book says that e = limn→∞1 +1nn. This means:(a) e is not really a number because it is a limit(b) e cannot be computed(c) the sequence of numbers21,322,433, ...,101100100, ... getas close as you want to the number e103. [P] When you read in the newspaper thing like inflation rate, interestrate, birth rate, etc., it always meansf0f, not f0itself.True or False.f0fis not the derivative of a
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