# How do you use the chain rule to find the derivative of log x?

Apr 3, 2015

I don't know if you are using $\log$ for ${\log}_{10}$ or for the natural log.

I would not use the chain rule. I define $\ln x$ as a definite integral, then define ${e}^{x}$ as the inverse function of $\ln x$. Then ${10}^{x} = {e}^{x \ln 10}$

If we define $y = \log x$ iff and only if ${10}^{y} = x$

It is not difficult to show that $\log x = \ln \frac{x}{\ln} 10$ so that $\frac{d}{\mathrm{dx}} \left(\log x\right) = \frac{1}{\ln} 10 \cdot \frac{1}{x}$ But I did not use the chain rule there.

Using $y = \log x$ iff and only if ${10}^{y} = x$ We might differentiate implicitly, which is a way of using the chain rule:

$\frac{d}{\mathrm{dx}} \left({10}^{y}\right) = \frac{d}{\mathrm{dx}} \left(x\right)$

${10}^{y} \ln 10 \frac{\mathrm{dy}}{\mathrm{dx}} = 1$

So, $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{{10}^{y} \ln 10} = \frac{1}{x} \ln 10$

If you're using $\log x$ for the inverse of ${e}^{x}$, then you could again use implicit differentiation, which is really the chain rule, to find the derivative.