Logarithmic Differentiation

[cv2yanks13]

Find the derivative of each of the functions listed below (with no negative exponents). You can leave all denominators in factored form (i.e you do not want to distribute the denominator)

a) ln(sqr(1+x)/(x^2+2)^3)

b) y= x^3(x-2)^4/sqr(x+1)

c) y=x^sinx

 

[BigGlenntheHeavy]

$\displaystyle What, \ did \ you \ have \ a \ sking \ accident? \ Are \ both \ of \ your \ arms \ in \ a \ sling?$

 

[cv2yanks13]

sorry.. I'm just having a really hard time understanding this... i'm looking for some help

 

[BigGlenntheHeavy]

$\displaystyle a) \ f(x) \ = \ ln\bigg[\frac{|1-x|^{1/2}}{(x^{2}+2)^{3}}\bigg] \ = \ ln|1-x|^{1/2}-ln(x^{2}+2)^{3}$

$\displaystyle = \ \frac{1}{2}ln|1-x|-3ln(x^{2}+2)$

$\displaystyle Hence, \ f'(x) \ = \ \frac{1}{2}\bigg(\frac{-1}{1-x}\bigg)-3\bigg(\frac{2x}{x^{2}+2}\bigg) \ = \ \frac{1}{2x-2}-\frac{6x}{x^{2}+2}$

$\displaystyle c) \ y \ = \ x^{sin(x)}$

$\displaystyle ln|y| \ = \ ln|x^{sin(x)}|, \ take \ log \ of \ both \ sides$

$\displaystyle ln|y| \ = \ sin(x)ln|x|, \ property \ of \ logs$

$\displaystyle \frac{y'}{y} \ = \ cos(x)ln|x|+sin(x)/x, \ implicit \ differentation$

$\displaystyle y' \ = \ y[cos(x)ln|x|+sin(x)/x]$

$\displaystyle y' \ = \ x^{sin(x)}[cos(x)ln|x|+sin(x)/x], \ resub.$

$\displaystyle Note: \ The \ sticking \ point \ here \ is \ one \ should \ have \ a \ good \ grasp \ of \ log \ properties.$