Derivative of $\ln\left(5 x\right)$

How to work it out

1. 1

   We differentiate $f(x) = \ln\left(5 x\right)$ with respect to x. The rule that applies is the chain rule.

   $$f(x) = \ln\left(5 x\right)$$
2. 2

   Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is $g(x) = 5 x$.

   $$g(x) = 5 x,\quad g'(x) = 5$$
3. 3

   The outer function is ln(u); its derivative evaluated at g(x), times g'(x):

   $$f'(x) = \left[\frac{1}{u}\right]_{u=g(x)} \cdot g'(x)$$
4. 4

   Simplify:

   $$f'(x) = \frac{5}{5 x}$$
5. 5

   Differentiating once more gives the second derivative, which tells you about curvature:

   $$f''(x) = \frac{-25}{\left(5 x\right)^{2}}$$

Values of f and f' for ln(5x)

Questions

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