Question

In Exercises \(31-42,\) find \(d y / d x\). $$y=\sqrt[3]{x}$$

Short answer

The derivative of \(y =\sqrt[3]{x}\) with respect to \(x\), \(dy/dx\), is \(dy/dx = \dfrac{1}{3} * x^{-2/3}\).

Step-by-step solution

Rewrite the function in power form

Before finding the derivative, we rewrite the cube root function, \(y = \sqrt[3]{x}\), in power form to simplify calculations. Therefore the function can be rewritten as \(y = x^{1/3}\).

Apply the power rule

We will apply the power rule of the derivative which states that if \(y = x^n\), then \(dy/dx = n*x^{(n-1)}\). Applying this rule to our function, \(dy/dx = (1/3) * x^{(1/3 - 1)}\).

Simplify the expression

Finally, we simplify our derivative expression. This yields \(dy/dx = (1/3) * x^{-2/3}\).

Key concepts

Power Rule

Understanding the power rule is essential for tackling calculus problems that involve differentiation of power functions. It's a basic rule used to find the derivative of a function where the variable is raised to a power. In algebraic terms, if we have a function like
\( y = x^n \), the power rule tells us that the derivative of this function with respect to \( x \) is
\( \frac{dy}{dx} = n \cdot x^{n-1} \).
The power rule makes finding derivatives for these types of functions a more straightforward process. In our exercise, \(y = \sqrt[3]{x}\) is first expressed as \( y = x^{1/3} \). Then, applying the power rule simplifies the differentiation to a manageable calculation.

Derivatives

The concept of derivatives is foundational in calculus. A derivative represents the rate at which a quantity changes. For example, if \(y\) changes with respect to \(x\), then the derivative
\( \frac{dy}{dx} \)
describes how much \(y\) changes for a small change in \(x\). In practical terms, it can represent things like the speed of a moving object or the slope of a function at any given point. While derivatives can seem complex at first, understanding how to use the rules of differentiation, like the power rule, makes it easier to calculate them for many different kinds of functions.

Algebraic Representation of Functions

In mathematics, expressing functions algebraically is a method of representing the relationships between variables. An algebraic function takes the form
\(y = f(x)\)
where \(y\) is the dependent variable and \(x\) is the independent variable. Different types of functions, such as linear, quadratic, or power functions, serve as the building blocks for more complex mathematical modeling. The cube root function in our example, \(y = \sqrt[3]{x}\), is better dealt with in its algebraic form \(y = x^{1/3}\) to facilitate the application of differentiation rules.

Simplifying Expressions

Simplifying expressions is a crucial process in mathematics that makes equations easier to work with. Simplification may involve combining like terms, using properties of exponents, or cancelling common factors. When dealing with derivatives, simplification helps to make the final derivative more understandable and easier to use in subsequent calculations or applications.
After applying the power rule
\( \frac{dy}{dx} = (1/3) \cdot x^{(1/3 - 1)} \),
we simplify the expression to
\( \frac{dy}{dx} = (1/3) \cdot x^{-2/3} \),
which translates to the derivative of \(y\) with respect to \(x\) when \(y\) is a cube root function of \(x\). The simplified form is not only neater but also more useful for further analyses, such as computing tangent lines or determining the behavior of the original function.