Question

How do you differentiate `y= ((x+3)(x^2+1)^3(x+1)^2)/(x^2+10)^(1/2)`?

Answer 1

`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))`

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 `ln(ab) = lna + lnb` and `ln(a/b) = lna - lnb`.

`lny = ln(x + 3) + ln(x^2 + 1)^3 + ln(x + 1)^2 - ln(x^2 +10)^(1/2)`

Now use `ln(a^n) = nlna`.

`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!