How do you find the derivative of the function y = log10(x)?

To find the derivative of the function y = log₁₀(x), we will use the rules of logarithmic differentiation. The logarithm with base 10, log₁₀(x), can be related to the natural logarithm using the change of base formula:

log₁₀(x) = log_e(x) / log_e(10)

Given that log₁₀(x) = (1 / log_e(10)) * log_e(x), we can rewrite the function as:

y = (1 / log_e(10)) * log_e(x)

Now, we can take the derivative with respect to x. The derivative of log_e(x) is:

dy/dx = (1 / (x * log_e(10)))

Therefore, the derivative of y = log₁₀(x) is:

dy/dx = 1 / (x * log_e(10))

In summary, the derivative of the function yields:

dy/dx = 1 / (x * log₁₀(e))

This formula provides a clear understanding of how fast log₁₀(x) changes with respect to x, showcasing the relationship between the natural logarithm and the logarithm with base 10.