Question

Evaluate \(\frac{d}{d x}\left(3^{x}\right)\).

Short answer

Answer: The derivative of the function \(f(x) = 3^x\) with respect to \(x\) is \(3^x \ln 3\).

Step-by-step solution

Identify the function and the derivative rule

First, let's identify the function and determine which derivative rule to use. In this case, we have a function in the form of \(f(x) = a^x\), where \(a = 3\). The rule for the derivative of an exponential function can be represented as: \(\frac{d}{d x}(a^x) = a^x \ln a\). Now we will apply this rule to our given function to find its derivative.

Apply the derivative rule to the function

Now that we have the correct derivative rule, let's apply it to our given function. Our given function is \(f(x) = 3^x\). So, the derivative of this function with respect to \(x\) is: \(\frac{d}{d x}(3^x) = 3^x \ln 3\).

Write down the answer

Now that we have applied the derivative rule to our given function, we found that the derivative of \(f(x) = 3^x\) with respect to \(x\) is: \(\frac{d}{d x}(3^x) = 3^x \ln 3\). So, the derivative of the given function is \(3^x \ln 3\).

Key concepts

Calculus

Calculus is an essential branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. As the study of change, calculus helps us understand the behavior of functions and provides tools for solving complex problems in engineering, science, economics, and beyond. Differentiation, one area of calculus, involves finding the derivative of a function, which measures how the function's output changes with respect to changes in the input. This concept is crucial for finding rates of change, optimizing processes, and understanding motion and growth patterns.

For instance, when evaluating the derivative of an exponential function like \(3^x\), we are seeking to understand how fast the value of the function grows or decays as \(x\) changes. Mastering differentiation and its rules, such as those for exponential functions, empowers students to tackle a wide range of practical and theoretical problems.

Exponential Function

An exponential function is of the form \(f(x) = a^x\), where \(a\) is a positive constant and is the base of the exponential. Unlike polynomial functions that grow at a rate proportional to their degree, exponential functions grow at a rate proportional to their current value, making them much faster speeding up as \(x\) increases. This property makes them incredibly important when modeling phenomena that multiply at constant rates, such as population growth, compound interest, and radioactive decay.

The base of an exponential can also be the mathematical constant \(e\), approximately 2.71828, which is known as Euler's number. The function \(e^x\) is particularly significant in calculus due to its unique properties, where the rate of growth of the function is equal to the value of the function at any point.

Natural Logarithm

The natural logarithm is the inverse function of the exponential function with the base of Euler's number \(e\). Denoted by \(\ln(x)\), the natural logarithm of a number represents the power to which \(e\) must be raised to obtain that number. It has distinctive properties that simplify the differentiation and integration of exponential functions. One of the most relevant properties, when dealing with derivatives, is that the derivative of \(\ln(x)\) with respect to \(x\) is \(1/x\), and the natural logarithm of \(e\) itself is 1.

In the context of the derivative of an exponential function with a base other than \(e\), we use the natural logarithm to find the constant factor that emerges after differentiation. As shown in the exercise, the derivative of \(3^x\) includes a factor of \(\ln(3)\), reflecting the relationship between the base of the function and the rate of change.

Derivative Rules

Derivative rules are formulas that provide a quick and direct method for finding the derivative of various functions without needing to apply the first principles of differentiation every time. Some of the most commonly used rules include the power rule, product rule, quotient rule, chain rule, and rules for differentiating trigonometric, logarithmic, and exponential functions.

The rule for derivative of an exponential function states that if \(f(x) = a^x\), then \(f'(x) = a^x \ln(a)\). This simplifies the differentiation process for any exponential function, as we can apply the rule directly to obtain the derivative, which reflects the function's inherent growth rate multiplied by the natural logarithm of its base, as seen in the exercise with the derivative of \(3^x\). Knowing how and when to apply these rules is necessary for effectively addressing calculus problems involving derivatives.