Questions: Differentiate. y=log10 x d/dx log10 x = □ □

Transcript text: Differentiate. \[ \begin{array}{r} y=\log _{10} x \\ \frac{d}{d x} \log _{10} x=\square \end{array} \] $\square$

Solution

Solution Steps

To differentiate the function $ y = \log_{10} x $, we can use the change of base formula for logarithms and the chain rule. The change of base formula allows us to express the logarithm in terms of the natural logarithm, and then we can differentiate using the known derivative of the natural logarithm.

Solution Approach

Use the change of base formula: $ \log_{10} x = \frac{\ln x}{\ln 10} $.
Differentiate $ \frac{\ln x}{\ln 10} $ with respect to $ x $.
Use the fact that the derivative of $ \ln x $ is $ \frac{1}{x} $.

Step 1: Change of Base Formula

We start with the function $ y = \log_{10} x $. Using the change of base formula, we can express this logarithm in terms of the natural logarithm: \[ y = \log_{10} x = \frac{\ln x}{\ln 10} \]

Step 2: Differentiate the Function

Next, we differentiate $ y $ with respect to $ x $: \[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{\ln x}{\ln 10} \right) \] Since $ \ln 10 $ is a constant, we can factor it out: \[ \frac{dy}{dx} = \frac{1}{\ln 10} \cdot \frac{d}{dx} (\ln x) \]

Step 3: Apply the Derivative of Natural Logarithm

The derivative of $ \ln x $ is $ \frac{1}{x} $: \[ \frac{dy}{dx} = \frac{1}{\ln 10} \cdot \frac{1}{x} = \frac{1}{x \ln 10} \]