Question

Determine the derivative of the given function and its domain. I have only one question about the domain.

Answer 1

The equation given is \[\ln (x ^{2}-18x)\]
I determined the derivative to be\[\frac{ 2(x-9) }{ x(x-18) }\]
Therefore my domain is determined by where the denominator equals zero, which it does at x=0 and x=18. In interval notation, I have \[(-\infty, 0) \cup (0,18) \cup (18, \infty)\]
I feel as if this is incorrect. Thoughts?

Answer 2

The question asked for the derivative of the given equation, and the domain of the Given Equation (not the domain of the derivative).

So in order to find the domain of the natural logarithm function, we need to know that it is undefined when the inside of the logarithm is zero or negative.

Thus we find the zeros of the equation:

\[x^2 -18x\]

Using the quadratic formula we find the zeros 0 and 18. (which makes sense since when the function is undefined, the derivative is also undefined at those points).

Now by analyzing the formula we can conclude that the equation is positive on these intervals:

]- infinity, 0[ U ]18, +infinity[,

and negative in between zero and 18. Thus the logarithm function is only defined on this interval:

\[)-\infty,0( \ U\ )18, +\infty(\]

Does that make sense?

-Dyl

Answer 3

Makes perfect sense. Interpreted the question wrong. Thanks.