Question

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=\ln 10^{x}$$

Short answer

\( \frac{dy}{dx}= \ln 10\)

Step-by-step solution

Rewrite the Equation

Start by rewriting the equation to make differentiation easier. Since \(\ln(a^b) = b \cdot \ln(a)\), we can rewrite the equation as \(y= x \ln 10\).

Differentiate

Next, differentiate both sides of the equation \(y= x \ln 10\) with respect to \(x\). The derivative of \(\ln 10\) is zero and the derivative of \(x\) is 1. Hence, \( \frac{dy}{dx}= \ln 10\).

Write The Final Answer

Finally, you can state that the derivative of \(y = \ln 10^{x}\) with respect to \(x\) is \( \frac{dy}{dx}= \ln 10\).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change (differential calculus) and the accumulation of quantities (integral calculus). It's a tool that allows us to understand the dynamics of motion and growth in natural processes, engineering, economics, and beyond.

When calculating derivatives, one of the fundamental operations in calculus, we frequently encounter functions involving exponents and logarithms. These functions can often be simplified before differentiation, which can be seen in the process of solving the given textbook exercise where we differentiate a logarithmic function.

Natural Logarithm Properties

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828. Understanding the properties of natural logarithms is crucial in calculus.

One key property, which is used in the exercise, is \(\ln(a^b) = b \cdot \ln(a)\). This property allows for simplification when dealing with the logarithms of exponential expressions. In the context of the problem, since logs and exponents are inverses, the simplification ultimately reveals the power of \(x\) as a coefficient, vastly simplifying the differentiation process.

Implicit Differentiation

Implicit differentiation is a powerful technique used in calculus when working with equations that define functions implicitly, rather than explicitly. This method involves differentiating both sides of an equation with respect to a variable and then solving for the derivative.

In the case of our textbook exercise, we employ a straightforward differentiation approach rather than implicit differentiation because the function is given explicitly as \(y = \ln 10^x\). However, knowing implicit differentiation is valuable when encountering more complex functions where expressing one variable explicitly in terms of another is not easily attainable.

Logarithmic Differentiation

Logarithmic differentiation is a specialized technique that simplifies the process of differentiating functions by taking the natural logarithm of both sides of an equation. This can be particularly useful when functions are products, quotients, or powers that would otherwise be more complex to differentiate using standard rules.

The technique wasn't necessary in our example since the function \(y = \ln 10^x\) was already simple enough to differentiate directly. Nevertheless, understanding how to apply logarithmic differentiation is valuable for tackling a broader range of problems involving more intricate functions.