Problem 16 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. $$ x^{3}-y^{2}=0 $$

Step-by-Step Solution

Answer

The derivative of the given function found by implicit differentiation is \(-\frac{3x^2}{2y}\)

Step 1: Differentiate the equation implicitly with respect to \(x\)

Firstly, apply the power rule to \(x^3\), which gives you \(3x^2\). Differentiate \(-y^2\) to get \(-2yy'\). Here, \(y'\) denotes \(dy/dx\). Therefore, the differentiated equation becomes \(3x^{2} -2yy' = 0\)

Step 2: Solve the equation for \(dy/dx\) or \(y'\)

Rearrange the equation to solve for \(y'\). Move \(2yy'\) to the other side and divide through by \(-2y\) which implies \(y' = -\frac{3x^2}{2y}\)

Step 3: Evaluate the derivative

The value of the derivative \(y'\) at any point can now be calculated by substituting the x and y coordinates of the point into the expression. The exact point was not provided in the task, for a general solution, the derivative is \(y' = -\frac{3x^2}{2y}\)

Key Concepts

Calculus

Calculus is a branch of mathematics that studies continuous change and is fundamental to many scientific disciplines. It is broadly classified into two areas: differential calculus, which deals with rates of change and slopes of curves; and integral calculus, which deals with accumulation of quantities and the areas under and between curves. In the context of the exercise, calculus is used to find the rate at which one quantity changes in relation to another, known as the derivative. Implicit differentiation is a technique within differential calculus used when a function is not given explicitly, but instead defined by an equation that relates several variables.

Derivative Calculation

Derivative calculation is a central operation in calculus that measures how a function changes as its input changes. Loosely speaking, a derivative represents the slope of the function at a particular point. For the given equation, \( x^3 - y^2 = 0 \), we calculate the derivative with respect to \( x \) using implicit differentiation because \( y \) is also a function of \( x \) but not given explicitly. After differentiating both sides of the equation with respect to \( x \) and applying appropriate rules of differentiation, we obtain a formula for \( dy/dx \) in terms of \( x \) and \( y \). This represents the rate at which \( y \) changes with respect to \( x \) at any point along the curve defined by the original equation.

Power Rule

The power rule is a basic differentiation rule that is used to find the derivative of a function of the form \( x^n \) where \( n \) is any real number. The rule states that the derivative of such a function is \( nx^{n-1} \). In our exercise, we apply the power rule to differentiate \( x^3 \) which results in \( 3x^2 \). However, since \( y \) is also a function of \( x \) and has an exponent, we also apply the power rule in conjunction with the chain rule, resulting in \( -2yy' \), where \( y' \) denotes \( dy/dx \) and is the derivative we are solving for.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They are frequently used to describe various phenomena such as physics, biology, economics, and engineering. The equation posed in the exercise, \( x^3 - y^2 = 0 \), can be viewed as a simple differential equation since it relates the variable \( x \) with the function \( y \) and its derivative \( dy/dx \). To solve it, we use implicit differentiation to find the relationship between \( dy/dx \) and the variables \( x \) and \( y \) without necessarily solving for \( y \) explicitly. This illustrates an elementary method of solving such equations when the function is given implicitly.