Derivative of $\ln\left(2 x\right)$

Q: What is the derivative of ln(2x)?

The derivative of ln ≤ft(2 x) is (2)/(2 x).

The derivative of $\ln\left(2 x\right)$ is $\frac{2}{2 x}$ .

f(x) =

\frac{d}{dx}\left[\ln\left(2 x\right)\right] = \frac{2}{2 x}

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\frac{d}{dx}\left[\ln\left(2 x\right)\right] = \frac{2}{2 x}

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Graph of f(x) = ln(2x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.

How to work it out

1
We differentiate $f(x) = \ln\left(2 x\right)$ with respect to x. The rule that applies is the chain rule.
$f(x) = \ln\left(2 x\right)$
2
Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is $g(x) = 2 x$ .
$g(x) = 2 x,\quad g'(x) = 2$
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
Simplify:
$f'(x) = \frac{2}{2 x}$
5
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = \frac{-4}{\left(2 x\right)^{2}}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[\ln\left(2 x\right)\right] = \frac{2}{2 x} \]
\[ f(x) = \ln\left(2 x\right) \]
\[ g(x) = 2 x,\quad g'(x) = 2 \]
\[ f'(x) = \left[\frac{1}{u}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{2}{2 x} \]
\[ f''(x) = \frac{-4}{\left(2 x\right)^{2}} \]

Values of f and f' for ln(2x)

x	f(x)	f'(x)
0.5	0	2
1	0.69314718	1
2	1.3862944	0.5
3	1.7917595	0.33333333

Questions

What is the derivative of ln(2x)?

The derivative of $\ln\left(2 x\right)$ is $\frac{2}{2 x}$ .

What is the second derivative of ln(2x)?

-4 / (2x)^2.

Which rule do you use for ln(2x)?

The chain rule.

What is the slope of ln(2x) at x = 1?

Derivative of ln⁡(2x)\ln\left(2 x\right)ln(2x)

How to work it out

Values of f and f' for ln(2x)

Questions

Related derivatives

Derivative of $\ln\left(2 x\right)$