10 Logarithmic Properties as Aids to Differentiation Differentiate:
11 Logarithmic Properties as Aids to Differentiation Differentiate:
12 Logarithmic Differentiation Differentiate:This can get messy with the quotient or product and chain rules. So we will use ln rules to help simplify this and apply implicit differentiation and then we solve for y’…
13 Derivative Involving Absolute Value Recall that the ln function is undefined for negative numbers, so we often see expressions of the form ln|u|. So the following theorem states that we can differentiate functions of the formy= ln|u| as if the absolute value symbol is not even there.If u is a differentiable function such that u≠0 then:
14 Derivative Involving Absolute Value Differentiate:
15 Finding Relative Extrema Locate the relative extrema of 𝑦=ln( 𝑥 2 +2𝑥+3)Differentiate:𝑢= 𝑥 2 +2𝑥+3, 𝑢 ′ =2𝑥+2𝑑𝑦 𝑑𝑥 = 2𝑥+2 𝑥 2 +2𝑥+3Set = 0 to find critical points2𝑥+2 𝑥 2 +2𝑥+3 =02x+2=0X=-1, Plug back into original to find yy=ln(1-2+3)=ln2 So, relative extrema is at (-1, ln2)
16 Homework5.1 Natural Logarithmic Functions and the Number e Derivative #19-35,47-65, 71,79,93-96
17 General Power Rule for Integration 𝑥 𝑛 𝑑𝑥= 𝑥 𝑛+1 𝑛+1 +𝑐, 𝑛≠−1Recall that it has an important disclaimer- it doesn’t apply when n = -1. So we can not integrate functions such as f(x)=1/x.So we use the Second FTC to DEFINE such a function.
18 Integration FormulasLet u be a differentiable function of x
33 Guidelines for integration Learn a basic list of integration formulas. (including those given in this section, you now have 12 formulas: the Power Rule, the Log Rule, and ten trigonometric rules. By the end of section 5.7 , this list will have expanded to 20 basic rules)Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula.If you cannot find a u-substitution that works, try altering the integrand. You might try a trigonometric identity, multiplication and division by the same quantity, or addition and subtraction of the same quantity. Be creative.If you have access to computer software that will find antiderivatives symbolically, use it.
34 Integrals of the Six Basic Trigonometric Functions
35 Homework5.2 Log Rule for Integration and Integrals for Trig Functions (substitution)#1-39, 47-53, 67