Question

T/F: Implicit differentiation can be used to find the derivative of \(y=x^{3 / 4}\).

Short answer

False: Implicit differentiation is unnecessary.

Step-by-step solution

Understanding Implicit Differentiation

Implicit differentiation is a method used when a function is not explicitly defined as one variable in terms of another. This means it is used for equations where variables are not separated, like those that define curves implicitly, i.e., equations like \( x^2 + y^2 = 1 \).

Given Function Analysis

The function given is \( y = x^{3/4} \). This function explicitly defines \( y \) as a function of \( x \). Therefore, \( y \) is independently expressed in terms of \( x \).

Choose Appropriate Differentiation Method

Since \( y \) is explicitly written as a function of \( x \), the conventional differentiation methods, like the power rule, are applicable instead. Implicit differentiation is unnecessary here.

Apply Power Rule

Differentiate the function using the power rule: if \( y = x^{n} \), then \( \frac{dy}{dx} = nx^{n-1} \). For \( y = x^{3/4} \), \( \frac{dy}{dx} = \frac{3}{4}x^{3/4 - 1} = \frac{3}{4}x^{-1/4} \).

Key concepts

Explicit Functions

In mathematics, explicit functions are equations where the dependent variable is isolated on one side of the equation. Simply put, these functions express one variable directly in terms of another. For example, the equation \(y = x^{3/4}\) is an explicit function because the dependent variable \(y\) is clearly expressed in terms of the independent variable \(x\). This clear expression allows us to apply different mathematical operations or calculus principles directly to find derivatives, evaluate functions, or perform other analyses. Explicit functions are straightforward because they show a direct relationship, making them easier to handle in calculus compared to implicit relationships. Understanding whether a function is explicit or implicit helps in choosing the right method for differentiation.

Power Rule

The power rule is a fundamental concept in differential calculus that provides a quick way to find the derivative of a function of the form \(y = x^{n}\), where \(n\) is any real number. According to the power rule, the derivative of \(y = x^{n}\) with respect to \(x\) is \(\frac{dy}{dx} = nx^{n-1}\).In our example, the function \(y = x^{3/4}\), the power rule tells us to differentiate as follows:

• Multiply by the exponent: \(n = \frac{3}{4}\), so the derivative will begin with \(\frac{3}{4}\).
• Subtract one from the exponent: \(3/4 - 1 = -1/4\).

Thus, the derivative is \(\frac{dy}{dx} = \frac{3}{4}x^{-1/4}\). The power rule simplifies the differentiation process and is one of the most utilized rules in calculus.

Conventional Differentiation Methods

Conventional differentiation methods include established techniques that simplify the process of finding derivatives for explicit functions. These methods, such as the power rule, product rule, or chain rule, are employed based on the structure of the given function or equation. In the case of the explicit function \(y = x^{3/4}\), using implicit differentiation would be unnecessary and inefficient. Instead, traditional approaches like the power rule are preferred due to their straightforward application. Conventional methods prioritize brevity and accuracy, minimizing the steps needed to obtain the derivative. Recognizing when a function is explicit allows you to apply these methods directly, making calculus not only approachable but also efficient and less error-prone.