# How do you differentiate f(x)=lnx+ln2x+3x using the sum rule?

Nov 20, 2015

$f ' \left(x\right) = \frac{2}{x} + 3$

#### Explanation:

The derivative of $\ln x$ is $\frac{1}{x}$.

To find the derivative of $\ln 2 x$ you'll need either the chain rule or the property of logarithms: $\ln 2 x = \ln 2 + \ln x$ and $\ln 2$ is a constant, so its derivative is $0$.
Therefore, $\frac{d}{\mathrm{dx}} \left(\ln 2 x\right) = \frac{d}{\mathrm{dx}} \left(\ln 2 + \ln x\right) = \frac{d}{\mathrm{dx}} \left(\ln 2\right) + \frac{d}{\mathrm{dx}} \left(\ln x\right) = 0 + \frac{1}{x} = \frac{1}{x}$.
(Note that this is a use of the sum rule.)

The derivative of $3 x$ is $3$.

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(\ln x\right) + \frac{d}{\mathrm{dx}} \left(\ln 2 x\right) + \frac{d}{\mathrm{dx}} \left(3 x\right)$
(This is the sum rule.)

$= \frac{1}{x} + \frac{1}{x} + 3 = \frac{2}{x} + 3$

If you prefer to write the derivative as a single ratio, get a common denominator and write:

$f ' \left(x\right) = \frac{2}{x} + \frac{3 x}{x} = \frac{2 + 3 x}{x} \text{ or } \frac{3 x + 2}{x}$