How do you differentiate #y= ((x+3)(x^2+1)^3(x+1)^2)/(x^2+10)^(1/2)#?
1 Answer
Explanation:
Use logarithmic differentiation.
#lny = ln(((x + 3)(x^2 + 1)^3(x +1)^2)/(x^2 + 10)^(1/2))#
Use the logarithm laws that state
#lny = ln(x + 3) + ln(x^2 + 1)^3 + ln(x + 1)^2 - ln(x^2 +10)^(1/2)#
Now use
#lny = ln(x + 3) + 3ln(x^2 + 1) + 2ln(x + 1) - 1/2ln(x^2 + 10)#
#1/y(dy/dx) = 1/(x +3) + (3(2x))/(x^2 + 1) + 2/(x + 1) - 1/2(2x)/(x^2 + 10)#
#1/y(dy/dx) = 1/(x + 3) + (6x)/(x^2 + 1) + 2/(x +1) - x/(x^2 + 10)#
#dy/dx= y(1/(x +3) + (6x)/(x^2 +1) + 2/(x + 1) - x/(x^2 + 10))#
#dy/dx= ((x + 3)(x^2 + 1)^3(x + 1)^2)/(x^2 + 10)^(1/2))(1/(x+ 3) + (6x)/(x^2 +1) + 2/(x + 1) - x/(x^2 + 10))#
Hopefully this helps!
