M408C: The Natural Logarithmic FunctionNovember 11, 2008We define the natural logarithm function byThe Fundamental Theorem of Calculus tells us what the derivative of ln x is:We can also prove the basic rules of the natural logarithm using calculus:It is helpful to know the behavior of ln x for graphing:We use the natural logarithm to define e:There are some additional formulas involving the natural logarithm you should know.Logarithmic differentiation is an extremely useful technique to differentiate some tricky functions:1.2.13. (7.2*.28) Differentiate y = ln(x4sin2x).Solution: Using the differentiation formulas for ln, we find:y0=1x4sin2xx4sin2x 0=1x4sin2x4x3· sin2x + x4· 2 sin x (sin x))0 =4x3sin2x + 2x4sin x cos xx4sin2x=4x+ 2 cot x4. (7.2*.60) Find the derivative of y =(x3+1)4sin2x3√x.Solution: We will use logarithmic differentiation. We begin by taking ln of both sides andsimplifying:ln y = ln(x3+ 1)4sin2x3√x= ln((x3+ 1)4) + ln(sin2x) − ln(x13)= 4 ln(x3+ 1) + 2 ln(sin x) −13ln(x)We then implicitly differentiate to get1yy0= 4 ·3x2x3+ 1+ 2 ·cos xsin x−13·1x.Finally, solving for y0and plugging in for y, we gety0= y12x2x3+ 1+ 2 cot x −13x=(x3+ 1)4sin2x3√x12x2x3+ 1+ 2 cot x −13x.5. (7.2*.72) Evaluate the integralRsin(ln x)xdx.Solution: Set u = ln x, so du =1xdx. ThusZsin(ln x)xdx =Zsin u du = − cos u + C = − cos(ln x) +
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