Problem 18 - The Calculus with Analytic Geometry

Question

Consider \(y\) as the independent variable and find \(D_{y} x\). $$ y=2 x^{3}-5 x $$

Step-by-Step Solution

Answer

The derivative of x with respect to y is \( D_y x = \frac{1}{6x^2 - 5} \).

Step 1 - Identify the given function

The given function is expressed as:y = 2x^3 - 5xThe goal is to find the derivative of x with respect to y, denoted as \( D_y x \).

Step 2 - Differentiate the function with respect to x

Differentiate both sides of the equation \( y = 2x^3 - 5x \) with respect to x:\( \frac{dy}{dx} = \frac{d}{dx} (2x^3 - 5x) \)Apply the power rule and sum rule of differentiation:\( \frac{dy}{dx} = 6x^2 - 5 \)

Step 3 - Use implicit differentiation to find \( D_y x \)

We need to find the derivative of x with respect to y. Therefore, take the reciprocal of \( \frac{dy}{dx} \):\( D_y x = \frac{dx}{dy} = \frac{1}{ \frac{dy}{dx} } \)

Step 4 - Substitute \( \frac{dy}{dx} \) with the expression

Substitute \( \frac{dy}{dx} \) from Step 2 into the equation:\( D_y x = \frac{1}{6x^2 - 5} \)

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique for finding the derivative of a function when it is not explicitly stated in terms of the independent variable. Unlike explicit differentiation where we directly differentiate \(y = f(x)\), implicit differentiation considers the relationship between the dependent and independent variables. We handle implicit functions by taking the derivative of both sides with respect to one variable (usually \(x\)) and then solving for the derivative we need. By treating all variables equally during differentiation and later isolating the desired derivative, we can effectively work with more complex relationships.

Reciprocal Rule

The reciprocal rule is a handy tool in differentiation, especially when we need to find the derivative with respect to a different variable. For example, if we have the derivative \(\frac{dy}{dx}\) and we need \(\frac{dx}{dy}\), we use the reciprocal rule. This states that \(\frac{dx}{dy}\) is the reciprocal of \(\frac{dy}{dx}\); in other words, \(\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}\). This approach is crucial when switching the roles of dependent and independent variables. In the given problem, once \(\frac{dy}{dx}\) is found, we directly take its reciprocal to get \(D_y x\), thus simplifying our process.

Power Rule

The power rule is one of the simplest and most widely used rules in differentiation. It states that if you have a function \(f(x) = x^n\), then its derivative is \(f'(x) = nx^{n-1}\). This rule makes it easy to find the derivatives of polynomial terms. For instance, in the function \(y = 2x^3 - 5x\), we apply the power rule to each term individually. For \(2x^3\), the derivative is \(6x^2\) (`2 \times 3x^{3-1}`), and for \(-5x\), the derivative is \(-5\) (`-5 \times 1x^{1-1}`). Combining these results, we get \(\frac{dy}{dx} = 6x^2 - 5\).

Inverse Function

Inverse functions reverse the roles of the dependent and independent variables. If \(f(x)\) is a function, its inverse, denoted as \(f^{-1}(x)\), gives us back the original value used in \(f(x)\). For differentiation, if \(y = f(x)\), then its derivative in terms of \(x\), \(\frac{dy}{dx}\), can be related to \(\frac{dx}{dy}\) through the reciprocal rule: \(\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}\). In practical terms, finding the derivative of an inverse function often involves both the original function and its derivative. By understanding these relationships, we can easily switch perspectives between the function and its inverse.