Question

How do I use logarithmic differentiation to determine the derivative for `f(x)=((x^5)((x-7)^5))/(x^2+2)^8`?

I saw something online saying I'm supposed to use In(x) and somehow take parts of this out using log? The example we did in class was like x^6 or something, and I have no clue how to use In(x) (if that's even what I need to use here) for an example so large. Thanks for your help!

Answer 1

Let `y = f(x)`.

`y=((x^5)((x-7)^5))/(x^2+2)^8`

Take the natural logarithm of both sides:

`ln(y)=ln(((x^5)((x-7)^5))/(x^2+2)^8)`

I am going to simplify this in slow motion.

Multiplication within the argument becomes the sum of two logarithms:

`ln(y)=ln(x^5)+ ln(((x-7)^5)/(x^2+2)^8)`

Division within the argument becomes the difference of two logarithms:

`ln(y)=ln(x^5)+ ln((x-7)^5)-ln((x^2+2)^8)`

The property `ln(a^c) = (c)ln(a)` allows further simplification:

`ln(y)=5ln(x)+ 5ln(x-7)-8ln(x^2+2)`

Differentiate both sides:

`1/y dy/dx=5/x+ 5/(x-7)-16x/(x^2+2)`

Multiply both sides by y:

`dy/dx=(5/x+ 5/(x-7)-16x/(x^2+2))y`

Substitute `y=((x^5)((x-7)^5))/(x^2+2)^8`:

`dy/dx=(5/x+ 5/(x-7)-16x/(x^2+2))((x^5)((x-7)^5))/(x^2+2)^8`

I will leave further simplification to you.