Answer: The derivative of log x is 1/(x ln 10)​.

To find this derivative, we use the differentiation rules for logarithmic functions. For the natural logarithm function, ln x (or loge​ x), where the base is the constant e, the derivative is:

d/dx​ (ln x) = 1/x​

This result comes from applying the chain rule of differentiation. The natural logarithm function represents the logarithm to the base e, approximately 2.71828.

So, the derivative of the natural logarithm function, log x, with respect to x is 1/x​, derived from the differentiation rules for logarithmic functions, where the derivative is inversely proportional to x.

Conclusion:

The derivative of the natural logarithm function, log(x), with respect to x is 1/(x ln 10), where ln 10 represents the natural logarithm of 10. This result is derived from the differentiation rules for logarithmic functions, specifically applying the chain rule of differentiation. The natural logarithm function, ln(x), represents the logarithm to the base e, approximately 2.71828

Some Related Questions:

How does the derivative of the natural logarithm function compare to other logarithmic functions?

The derivative of the natural logarithm function, ln(x), is 1/x, which differs from the derivatives of logarithmic functions with bases other than e. Each logarithmic function has its derivative based on its base and follows specific rules of differentiation.

Can you explain the significance of ln 10 in the derivative of log(x)?

ln 10 represents the natural logarithm of the number 10, which is approximately 2.30259. It appears in the derivative of log(x) as a constant factor, affecting the rate of change of the function.

How is the derivative of log(x) used in practical applications?

The derivative of log(x) has applications in various fields such as finance, biology, and physics. It is used to analyze rates of growth or decay, determine exponential functions’ behavior, and model processes involving logarithmic relationships.