Question

Calculate the derivative of the following functions. $$y=\log _{10} x$$

Short answer

Answer: The derivative of the function \(y=\log _{10} x\) is \(\frac{dy}{dx} = \frac{1}{x \ln 10}\).

Step-by-step solution

Rewrite in natural logarithm format

We need to calculate the derivative of the function \(y=\log _{10} x\). Since the function is in the base 10 logarithm, we should rewrite it in terms of the natural logarithm (base \(e\)) to simplify further calculations. Using the change of base formula: $$y=\frac{\ln x}{\ln 10}$$

Differentiate the function

Now, find the derivative of the rewritten function with respect to x. Apply the chain rule. To apply the chain rule, let \(u(x) = \ln x\) and \(v(x) = \ln 10\). The derivative of \(y\) with respect to \(x\) is: $$\frac{dy}{dx}=\frac{d(u/v)}{dx}$$ Since \(\ln 10\) is a constant value, we have: $$\frac{d(u/v)}{dx}=\frac{d(u\cdot (1/v))}{dx} = v^{-1}\cdot \frac{du}{dx} = \frac{1}{\ln 10} \cdot \frac{d(\ln x)}{dx}$$

Apply the differentiation formula for natural logarithm

Now, apply the differentiation formula for the natural logarithm function, which is: $$\frac{d(\ln x)}{dx} = \frac{1}{x}$$

Combine the results

Combine the results from Steps 2 and 3: $$\frac{dy}{dx} = \frac{1}{\ln 10} \cdot \frac{d(\ln x)}{dx} = \frac{1}{\ln 10} \cdot \frac{1}{x}$$ Now, we have found the derivative of the function: $$\frac{dy}{dx} = \frac{1}{x \ln 10}$$

Key concepts

Change of Base Formula

When dealing with logarithms, it's often useful to convert one base to another. This is particularly true when differentiating functions involving logarithms with bases other than the natural base, which is \(e\). The change of base formula is a helpful tool in these scenarios. It allows for conversion of logs from any base to the natural logarithm (base \(e\)), making calculus operations more straightforward.

The change of base formula for logarithms is given by:

• \( \log_b{x} = \frac{\ln x}{\ln b} \)

In this formula, \(b\) represents the original base you want to change, while \(x\) is the argument of the logarithm. This formula essentially states that you can take the natural log of the argument divided by the natural log of the base to convert it.

In our exercise, the logarithmic function \(y = \log_{10} x\) begins as a base 10 logarithm and is transformed using the change of base formula. Thus, \(y\) is rewritten as:

• \( y = \frac{\ln x}{\ln 10} \)

By using this conversion, the problem shifts to working with a natural logarithm, which is more convenient for calculus applications.

Differentiation Using Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. When you have a function composed of two or more functions, the chain rule helps to break it down, allowing you to differentiate effectively. In essence, the chain rule states that:

• If you have a composite function \(h(x) = f(g(x))\), then its derivative \(h'(x)\) is \(f'(g(x)) \cdot g'(x)\).

In our context, where we have \(y = \frac{\ln x}{\ln 10}\), this involves applying the chain rule due to the division of two functions. Recognizing that \(\ln 10\) is a constant, we apply the derivative rules and chain rule as follows:

• Take the derivative of the natural log part \(\ln x\), resulting in \(\frac{1}{x}\).
• Combine with the constant multiplier \(\frac{1}{\ln 10}\) to obtain \(\frac{1}{\ln 10} \cdot \frac{1}{x}\).

These steps show the application of the chain rule, providing the gradient of the logarithmic function in respect to \(x\). Each component of the function is differentiated separately, then combined using multiplication to yield the final result.

Derivative of Natural Logarithm

The natural logarithm function \(\ln x\) is quite significant in calculus due to its simple differentiation rule. Differentiating \(\ln x\) yields a result that's particularly elegant and easy to use in further calculations. The rule for differentiation of the natural logarithm is:

• \(\frac{d}{dx}(\ln x) = \frac{1}{x}\)

This derivative tells us that the rate of change of \(\ln x\) with respect to \(x\) is the reciprocal of \(x\). This simplicity makes it easy to apply this rule to various contexts, including the logarithmic function from the exercise.

In our problem, once \(y\) is transformed to \(\frac{\ln x}{\ln 10}\), we rely on this straightforward differentiation rule for \(\ln x\) to evaluate the overall derivative. Employing this rule results in a clear and concise step by step differentiation process, culminating in the solution:

• \(\frac{dy}{dx} = \frac{1}{x \ln 10}\)

Understanding this derivative rule for the natural log is crucial, as it readily applies to complex functions involving logarithmic expressions, simplifying the differentiation process considerably.