Question

Find `(dy)/(dx)` of `y=((x+1)^4(x-5)^3)/(x-3)^8`?

Answer 1

`dy/dx = (((x + 1)^4(x - 5)^3)/(x- 3)^8)(4/(x + 1) +3/(x- 5) - 8/(x- 3))`

Explanation:

Use logarithmic differentiation.

`lny = ln(((x + 1)^4(x - 5)^3)/(x - 3)^8)`

`lny = ln(x +1)^4 + ln(x -5)^3 - ln(x - 3)^8`

`lny = 4ln(x + 1) + 3ln(x - 5) - 8 ln(x - 3)`

Now take the derivative of both sides.

`1/y(dy/dx) = 4/(x + 1) + 3/(x - 5) - 8/(x -3)`

`dy/dx= y(4/(x + 1) + 3/(x - 5) - 8/(x- 3))`

`dy/dx = (((x + 1)^4(x - 5)^3)/(x- 3)^8)(4/(x + 1) +3/(x- 5) - 8/(x- 3))`

Hopefully this helps!