So I think I might be going about the chain rule in the wrong way.
Currently this is how I do the chain rule:
$ \frac {d}{dx} (x^2 + 4)^2 $ I would carry down the two like normal using the power rule but since this is a composition of function I have to use the chain rule so I use that factor times the inner function and then times the derivative of the inner function. I get $2(x^2 + 4)2x$.
But now I am a little confused with logarithmic functions, I know how to do them but I am just not sure why and the book is no help.
For something like the derivative of $ \log_{10} (x^3 + 1)$ I know using the library I have $\frac {d}{dx} (\log_a x) = \frac {1}{x\ln a}$ so for this problem I am left with the derivative of the function times the derivative of the inner function.
I get $\frac {3x^2}{(x^3 + 1)\ln10}$
I know this is the correct answer, but my logic is not correct. Why does it seem like I do partial derivatives for things like the first example but then I do full derivatives for logarithms. What am I missing?