Problem 13 - Calculus Early Transcendentals

Question

Find the derivative of the following functions. See Example 4 of Section 3.1 for the derivative of \(\sqrt{x}\). $$f(x)=5 x^{3}$$

Step-by-Step Solution

Answer

Answer: The derivative of the function \(f(x) = 5x^3\) is \(f'(x) = 15x^2\).

Step 1: Identify the function and its components

The given function is: $$f(x) = 5x^3$$ Here, the value of \(a=5\) and \(n=3\).

Step 2: Apply the power rule

Now, we will apply the power rule to the function \(f(x) = 5x^3\). For the function \(ax^n\), the derivative is given by: $$\frac{d}{dx}(ax^n) = n(ax^{n-1})$$ So, for our function: $$\frac{d}{dx}(5x^3) = 3(5x^{3-1})$$

Step 3: Simplify the derivative

Next, we simplify the derivative as follows: $$f'(x) = 3(5x^{2})$$ $$f'(x) = 15x^2$$ The derivative of the function \(f(x) = 5x^3\) is: $$f'(x) = 15x^2$$

Key Concepts

Power Rule

Understanding the power rule is essential when taking the derivative of functions in calculus, especially when dealing with polynomial terms like those found in the function f(x) = 5x^3. The power rule is a straightforward but mighty tool, stating that if you have a function of the form ax^n, where a is a constant and x is raised to an exponent n, the derivative of this function with respect to x is anx^{(n-1)}.

When applying the power rule, you are essentially reducing the exponent by one and multiplying by the original exponent. This method streamlines the process of finding the derivative and can be quickly applied to each term individually when working with polynomials. It's important to remember that the power rule can only be used when the exponent is a real number, which makes it especially suited for functions like the one in our example.

Simplifying Derivatives

Simplifying derivatives is a vital step in calculus as it helps to make complex expressions more manageable and easier to work with. After applying the power rule to our function f(x) = 5x^3, we get the derivative f'(x) = 3(5x^2). To simplify this expression, we follow basic arithmetic and algebra rules to combine like terms and constants.

For this derivative, we multiply the constant outside the parenthesis by each term inside. In this case, 3 multiplied by 5 results in 15, and the x^2 remains unchanged. Thus, the significantly simpler and final form of the derivative is f'(x) = 15x^2. Simplification is critical for making further calculations easier, such as evaluating the derivative at specific points or integrating the function.

Calculus

Calculus, the branch of mathematics that studies the changes between values, is perfectly exemplified by the concept of derivatives. The derivative is a tool that allows us to determine the instantaneous rate of change of a function, or, in simpler terms, how a function is changing at any given point along its curve.

The derivative of a function at a point is the slope of the tangent line to the function's graph at that point. This is analogous to finding the speed of an object at a specific moment in time in a physical context. In the example with f(x) = 5x^3, calculating the derivative gives us a formula f'(x) = 15x^2 which can predict how quickly the values of f(x) are changing in relation to changes in x. This fundamental concept is not only foundational in mathematics but also across many scientific disciplines including physics, engineering, economics, and beyond, showcasing the universal importance of calculus.