Question

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

Short answer

False, implicit differentiation is not needed for \(y=\sqrt{x}\).

Step-by-step solution

Understand Implicit Differentiation

Implicit differentiation is a technique used when a function cannot be easily solved for one variable, usually because it is presented in an implicit form of the equation involving two variables.

Analyze Given Equation

The equation provided is in explicit form: \(y = \sqrt{x}\). This is already solved for \(y\) in terms of \(x\), and it does not require implicit differentiation since \(y\) is isolated.

Determine Applicability

Implicit differentiation is primarily used for functions where both variables are mixed on one side of the equation, such as equations like \(x^2 + y^2 = 1\). Since \(y = \sqrt{x}\) is explicit, we can directly differentiate without requiring implicit differentiation.

Conclusion on Implicit Differentiability

Since \(y = \sqrt{x}\) is already explicit, the true statement is that implicit differentiation is not necessary to find the derivative. We can use standard differentiation techniques.

Key concepts

explicit form

An equation is said to be in explicit form when one variable is expressed explicitly in terms of another variable. In other words, it is directly solved for one variable. For instance, in the equation \( y = \sqrt{x} \), the variable \( y \) is isolated and expressed as a function of \( x \). Thus, it is in explicit form.
Explicit form equations are straightforward and convenient because they allow for easier manipulation and differentiation.
In the context of differentiation, having an explicit form means you can apply basic rules of differentiation without additional steps. There’s no need for complex techniques like implicit differentiation.
For students familiar with calculus, spotting equations that are in explicit form can save time and effort when finding derivatives since these equations are directly formulated to apply basic differentiation rules.

derivative

The derivative of a function measures how the function's output value changes as the input value changes. In simple terms, it is a kind of mathematical "rate of change."
For an equation in explicit form, such as \( y = \sqrt{x} \), finding the derivative involves applying differentiation rules directly. In this example, the derivative, denoted \( \frac{dy}{dx} \), represents how \( y \) changes with respect to \( x \).
The process involves applying the power rule, which is one of the basic techniques in calculus for finding derivatives:

• The square root \( \sqrt{x} \) is rewritten as \( x^{1/2} \).
• The power rule states that \( \frac{d}{dx}(x^n) = nx^{n-1} \).
• Using the power rule, the derivative of \( x^{1/2} \) becomes \( \frac{1}{2}x^{-1/2} \).

Thus, the derivative of \( y = \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \). The derivative provides valuable insights, especially in fields like physics, economics, and engineering, where rate of change is crucial.

differentiation techniques

Differentiation techniques refer to the various methods used to compute the derivative of a function. These techniques can become necessary depending on the form of the equation.
While explicit forms like \( y = \sqrt{x} \) are direct and require basic differentiation rules, we sometimes encounter more complex equations. These situations call for specialized techniques, such as:

• Implicit Differentiation: Used when both variables \( x \) and \( y \) are intertwined and not explicitly solved for one another. It applies when equations are in implicit form, like \( x^2 + y^2 = 1 \).
• Chain Rule: Utilized for composite functions to differentiate the "outer" function and the "inner" function separately.
• Product and Quotient Rules: Necessary when dealing with functions that are products or quotients of two or more functions.

Differentiation techniques are essential tools in calculus, helping solve real-world problems where functions describe relationships between varying quantities. Understanding when and how to apply each technique can simplify the process of finding derivatives, ensuring correct and efficient problem-solving. For equations like \( y = \sqrt{x} \), basic differentiation techniques suffice, leading to straightforward solutions.