Differentiate $y=\log _{10} x$
Answer (1)
Rewrite the base-10 logarithm in terms of the natural logarithm using the change of base formula: lo g 10 x = l n 10 l n x .
Differentiate the expression with respect to x , treating l n 10 1 as a constant: d x d ( l n 10 l n x ) = l n 10 1 d x d ( ln x ) .
Apply the derivative of the natural logarithm: d x d ( ln x ) = x 1 .
Obtain the final derivative: d x d y = x l n 10 1 .
x ln 10 1
Explanation
Problem Setup We are given the function y = lo g 10 x and asked to find its derivative.
Change of Base To differentiate y = lo g 10 x , we first convert it to natural logarithm using the change of base formula:
Rewriting the Function lo g 10 x = l n 10 l n x
Differentiation Now, we differentiate with respect to x . Since l n 10 1 is a constant, we have:
Applying Constant Multiple Rule d x d y = d x d ( l n 10 l n x ) = l n 10 1 ⋅ d x d ( ln x )
Derivative of Natural Logarithm We know that the derivative of ln x is x 1 , so:
Final Derivative d x d y = l n 10 1 ⋅ x 1 = x l n 10 1
Conclusion Thus, the derivative of y = lo g 10 x is x l n 10 1 .
Examples
Logarithmic differentiation is used extensively in fields like finance to analyze growth rates and sensitivities. For example, if you have a function describing the value of an investment over time, taking the logarithm and then differentiating can simplify the analysis of the investment's rate of return. This approach transforms complex multiplicative relationships into simpler additive ones, making calculations and interpretations easier.
