To find the derivative of the function y = log₁₀(x), we can use the change of base formula for logarithms. The logarithm to base 10 can be expressed in terms of the natural logarithm:
log₁₀(x) = rac{ln(x)}{ln(10)}
Here, ln(x) is the natural logarithm of x.
Now, we can differentiate y with respect to x:
y’ = rac{d}{dx} igg( rac{ln(x)}{ln(10)} igg)
Since ln(10) is a constant, we can factor it out of the derivative:
y’ = rac{1}{ln(10)} rac{d}{dx} (ln(x))
The derivative of ln(x) is known to be rac{1}{x}. Therefore, we have:
y’ = rac{1}{ln(10)} imes rac{1}{x}
Finally, we can simplify this to:
y’ = rac{1}{x imes ln(10)}
This gives us the derivative of y = log₁₀(x) with respect to x.