What is the differentiation of log 2x ?

To find the derivative of the function \( \log(2x) \), we can apply the rules of logarithmic differentiation. The key here is to remember that the logarithm of a product can be simplified using properties of logarithms. Let's break it down step by step.

Understanding Logarithmic Differentiation

The function we are differentiating is \( \log(2x) \). Using the property of logarithms, we can rewrite this as:

• \( \log(2x) = \log(2) + \log(x) \)

This simplification is crucial because it allows us to differentiate each term separately. The logarithm of a constant, \( \log(2) \), will have a derivative of zero, while the derivative of \( \log(x) \) is well-known.

Calculating the Derivative

Now, let's differentiate:

• The derivative of \( \log(2) \) is \( 0 \).
• The derivative of \( \log(x) \) is \( \frac{1}{x} \).

Putting it all together, the derivative of \( \log(2x) \) is:

• \( \frac{d}{dx}[\log(2x)] = 0 + \frac{1}{x} = \frac{1}{x} \)

Applying the Chain Rule

Alternatively, if we want to differentiate \( \log(2x) \) directly using the chain rule, we can do it this way:

• Let \( u = 2x \). Then, \( \log(2x) = \log(u) \).
• Using the chain rule, we have \( \frac{d}{dx}[\log(u)] = \frac{1}{u} \cdot \frac{du}{dx} \).

Now, we find \( \frac{du}{dx} \) where \( u = 2x \):

• \( \frac{du}{dx} = 2 \)

Substituting back, we get:

• \( \frac{d}{dx}[\log(2x)] = \frac{1}{2x} \cdot 2 = \frac{1}{x} \)

Final Result

In both methods, we arrive at the same conclusion. The derivative of \( \log(2x) \) is:

\( \frac{1}{x} \)

This result shows that the rate of change of the logarithmic function \( \log(2x) \) with respect to \( x \) is inversely proportional to \( x \). Understanding these properties not only helps in calculus but also in various applications across mathematics and science.