Find the derivative below. Give both the domain and the range.

Question

Find the derivative below. Give both the domain and the range.

 

??/???? ln2??^8

Answer 1

When we differentiate something like ln(2x⁸), the easiest path is to simplify the log first, before taking any derivative at all. Using the log rules, the product inside splits apart and the exponent comes down in front, so it becomes:

ln(2x⁸) = ln 2 + 8 ln|x|

One thing to be careful about is the absolute value. Since x⁸ is positive for every x except 0, the original function is defined for negative x too, so writing 8 ln(x) would quietly throw away half of the domain.

Now the derivative is quick. ln 2 is just a constant, so its derivative is 0, and the derivative of ln|x| is 1/x, which gives us:

d/dx ln(2x⁸) = 8/x

If you notice, the 2 disappeared completely. That happens because a constant multiplied inside a log becomes a constant added outside of it, and constants die when we differentiate. The 8 is the only part of that exponent that survives.

For the domain, we need 2x⁸ > 0, which is true for every x except x = 0, and 8/x is undefined at 0 as well, so the domain is (−∞, 0) ∪ (0, ∞). For the range, 8/x can get as close to 0 as we want but never actually reaches it, and it does hit every other real number, so the range is (−∞, 0) ∪ (0, ∞) too.

So, our final answer is 8/x, with the domain and the range both being all real numbers except 0.