Problem 81 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \begin{equation}y=\log _{10} e^{x}\end{equation}

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{\ln(10)} \).

Step 1: Simplify Using Logarithmic Identity

Recall that the logarithm of an expression to any base is equivalent to the natural logarithm of the expression divided by the natural logarithm of the base. Thus, we can write: \[ y = \frac{\ln(e^x)}{\ln(10)} \]. Since \(\ln(e^x)\) equals \(x\) (because the natural logarithm of \(e^x\) is \(x\)), this simplifies to \[ y = \frac{x}{\ln(10)} \].

Step 2: Differentiate Simplified Expression

Now, with \( y = \frac{x}{\ln(10)} \), we differentiate with respect to \( x \). Constant multiple rule allows us to take the derivative directly: \[ \frac{dy}{dx} = \frac{1}{\ln(10)} \frac{d}{dx}x = \frac{1}{\ln(10)} \cdot 1 = \frac{1}{\ln(10)} \]. So, the derivative is \( \frac{1}{\ln(10)} \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used to find the derivatives of complex functions that are not easily differentiated using basic rules. This method leverages the properties of logarithms and the chain rule in calculus, making differentiation much simpler in certain situations.

To use logarithmic differentiation, follow these general steps:

• Take the natural log of both sides of the equation you want to differentiate. This allows you to simplify the expression.
• Use the properties of logarithms to break apart complex products and quotients into sums and differences, making the function easier to manage.
• Differentiate both sides of the equation. On the side with the original expression, use the chain rule.
• After finding the derivative, solve for the derivative of the original function.

In the context of the given problem, using a logarithmic identity made it easy to convert the original function into a form where basic differentiation rules could be applied directly.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a logarithm with base \( e \), where \( e \approx 2.71828 \), and it plays a crucial role in calculus. It's often used due to its mathematical properties, especially in calculus, where it simplifies many types of expressions and equations.

When working with exponential functions, like \( e^x \), the natural logarithm is particularly useful because it transforms these into linear functions, which are much easier to differentiate. For example, \( \ln(e^x) = x \), drastically simplifying the differentiation process.

Additionally, the natural logarithm is essential in solving equations involving exponential growth and decay, which are common in calculus problems across various scientific fields.

Calculus

Calculus is the branch of mathematics that studies how things change. It's split into two main areas: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which indicates how a function changes as its input changes, essentially looking at the rate of change.

In the original exercise, the task was to find the derivative of the function \( y = \log _{10} e^{x} \). This requires an understanding of basic differentiation rules and techniques, such as the constant multiple rule and the understanding of logarithmic functions.

Calculus is applicable across several domains, including physics, engineering, economics, statistics, and more, providing tools to model and solve problems involving dynamic systems. It helps us understand things like motion, growth, and area under curves through the use of derivatives and integrals.