derivate this:

Question

ln((x^2+x)^2/sqrt(x+1) 

Answer 1

ln[(x² + x)²/√(x + 1)]

= ln[(x² + x)²/(x + 1)^1/2]

{Apply the quotient rule in logarithm rule : ln(a/b) = lna - lnb.}

= ln(x² + x)² - ln(x + 1)^1/2

= 2ln(x² + x) +  ln(x + 1)/2

Differentiate with respect to x.

Apply the formula d/dx(lnx) = 1/x

d/dx(x)ⁿ = x^n-1

= 2 (2x + 1)/(x² + x)  - 1/2(x + 1)

= (4x + 2)/(x² + x) - 1/(2x + 2)

= [ (4x + 2)(2x + 2) - (x² + x)]/[(x² + x)(2x + 2)

= [ 8x² + 8x + 4x + 4 - x² - x]/[2x³ + 2x² + 2x² + 2x]

= (7x² + 11x + 4)/(2x³ + 2x² + 2x² + 2x)

= (7x² + 7x + 4x + 4)/(2x³ + 4x² + 2x)

= [7x(x + 1)+ 4(x + 1)]/2x(x² + 4 + 2)

= [ (7x + 4)(x + 1)/(2x(x + 1)²

= (7x+4)/2x(x + 1)

= (7x + 4)/(2x² + 2x).