How do you find the second derivative of # ln(x^2+5x)# ?

Question

Ezekiel Alexander · Accepted Answer

#-(2x^2+10x+25)/(x^2+5x)^2#

First, we need to know the derivative of #ln(x)#.

#d/dxln(x)=1/x#

So, since we have a function embedded within #ln(x)#, we must use the chain rule to differentiate it.

#d/dxln(f(x))=1/f(x)*f'(x)#

Thus:

#d/dxln(x^2+5x)=1/(x^2+5x)*d/dx(x^2+5x)#

#=(2x+5)/(x^2+5x)#

Now, to differentiate this again, use the quotient rule:

#d/dxf(x)/(g(x))=(f'(x)g(x)-f(x)g'(x))/(g(x))^2#

Thus:

#d^2/dx^2ln(x^2+5x)=((x^2+5x)d/dx(2x+5)-(2x+5)d/dx(x^2+5x))/(x^2+5x)^2#

#=((x^2+5x)(2)-(2x+5)(2x+5))/(x^2+5x)^2#

#=(2x^2+10x-(4x^2+20x+25))/(x^2+5x)^2#

#=(-2x^2-10x-25)/(x^2+5x)^2#

William Abdalla · Answer

#-(2x^2+10x+25)/(x^2+5x)^2# First derivative: #(1/(x^2+5x))*(2x+5)=(2x+5)/(x^2+5x)# Second derivftive: #(2(x^2+5x)-(2x+5)(2x+5))/(x^2+5x)^2# #=-(2x^2+10x+25)/(x^2+5x)^2#

Scarlett Allison · Answer

#(-2x^2-10x-25)/(x^2+5x)^2#

To find the first derivative use the #color(blue)"chain rule"#

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|)))..... (A)#

let #u=x^2+5xrArr(du)/(dx)=2x+5#

then #y=lnurArr(dy)/(du)=1/u#

substitute these values into (A) changing u back to x.

#rArrdy/dx=1/uxx(2x+5)=(2x+5)/(x^2+5x)# #color(blue)"-----------------------------------------------------"#

To find the second derivative use the #color(blue)"quotient rule"#

If #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)|)))#

here #g(x)=2x+5rArrg'(x)=2#

and #h(x)=x^2+5xrArrh'(x)=2x+5#

#f'(x)=((x^2+5x).2-(2x+5)(2x+5))/(x^2+5x)^2#

simplifying the numerator.

#=(2x^2+10x-4x^2-20x-25)/(x^2+5x)^2#

The second derivative is therefore.

#=(-2x^2-10x-25)/(x^2+5x)^2=(-(2x^2+10x+25))/(x^2+5x)^2#

Cameron Alexander · Answer

undefined To find the second derivative of ln(x^2 + 5x), follow these steps:

1. Find the first derivative of ln(x^2 + 5x) with respect to x.
2. Apply the chain rule if necessary.
3. Once you have the first derivative, differentiate it again with respect to x to find the second derivative.

Let's start by finding the first derivative:

dy/dx = d/dx [ln(x^2 + 5x)]

Now, applying the chain rule:

dy/dx = (1 / (x^2 + 5x)) * d/dx [x^2 + 5x]

Simplify the expression:

dy/dx = (1 / (x^2 + 5x)) * (2x + 5)

Now, differentiate again to find the second derivative:

d^2y/dx^2 = d/dx [(1 / (x^2 + 5x)) * (2x + 5)]

Apply the product rule and simplify:

d^2y/dx^2 = [(2x + 5) * d/dx(1 / (x^2 + 5x))] + [(1 / (x^2 + 5x)) * d/dx(2x + 5)]

Differentiate both terms separately:

d/dx(1 / (x^2 + 5x)) = -1 / (x^2 + 5x)^2 * d/dx(x^2 + 5x)

d/dx(2x + 5) = 2

Plug these values back into the second derivative expression:

d^2y/dx^2 = [(2x + 5) * (-1 / (x^2 + 5x)^2 * (2x + 5))] + [(1 / (x^2 + 5x)) * 2]

Simplify the expression further:

d^2y/dx^2 = [-2(2x + 5) / (x^2 + 5x)^2] + [2 / (x^2 + 5x)]

Thus, the second derivative of ln(x^2 + 5x) is:

d^2y/dx^2 = [-2(2x + 5) / (x^2 + 5x)^2] + [2 / (x^2 + 5x)]