How do you find the first and second derivative of #ln(x^2-4)#?
1 Answer
Oct 2, 2016
Explanation:
let
# f(x)=ln(x^2-4)#
#color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(d/dx(lnx)=1/x)color(white)(a/a)|)))# and more generally.
#color(red)(bar(ul(|color(white)(a/a)color(black)(d/dx(ln(g(x)))=(g'(x))/g(x))color(white)(a/a)|)))#
#rArrf'(x)=(2x)/(x^2-4)# To find the second derivative, use the
#color(blue)"quotient rule"# Given
#f(x)=g(x)/(h(x))" then"#
#color(red)(bar(ul(|color(white)(a/a)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(a/a)|)))......(A)# here
#g(x)=2xrArrg'(x)=2# and
#h(x)=x^2-4rArrh'(x)=2x# substitute these values into (A)
#f''(x)=((x^2-4).2-2x.2x)/(x^2-4)^2#
#=(2x^2-8-4x^2)/(x^2-4)^2=(-2x^2-8)/(x^2-4)^2=(-2(x^2+4))/(x^2-4)^2#
