Question

In Exercises, find \(d y / d x\) implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating \(d y / d x\) at the point. $$ 4 y^{2}-x^{2}=7 $$

Short answer

The derivative of the function, found both implicitly and explicitly, is \(\frac{dy}{dx} = \frac{x}{4y}\). This result proves the equivalence of the two methods. The slope of the tangent line at a specific point on the graph can be calculated by substituting the x-value of the point into the derivative expression.

Step-by-step solution

Implicit Differentiation

To take the derivative implicitly, apply the power rule and the chain rule. Start by taking the derivative of both sides of the equation with respect to \(x\). We obtain the following:\[2 * 4y * \frac{dy}{dx} - 2x = 0\]Simplify this to:\[8y * \frac{dy}{dx} = 2x\]Finally, solve for \(\frac{dy}{dx}\) to get:\[\frac{dy}{dx} = \frac{x}{4y}\]

Finding Explicit Function and its Derivative

The explicit function can be obtained by isolating \(y\) in the original equation:\[y = \sqrt{(x^2 + 7) / 4}\]Take the derivative of \(y\) with respect to \(x\) using chain rule, which results in:\[\frac{dy}{dx} = \frac{1}{2} * (\frac{x^2 + 7}{4})^{-1/2} * 2x\]Simplify this to:\[\frac{dy}{dx} = \frac{x}{4\sqrt{(x^2 + 7)/4}}\]The denominator of this result matches with the explicit function \(y\), so the derivative can be expressed as:\[\frac{dy}{dx} = \frac{x}{4y}\]

Verification of results

From the above two methods, the derivatives obtained are equivalent as \(\frac{dy}{dx} = \frac{x}{4y}\). Hence, the results confirm the equivalence of the two methods.

Evaluation of the Derivative at a Specific Point

To estimate the slope of the tangent line at a specific point on the graph of the function, substitute the x-value of the point into the derivative expression. Let's take a point \((a, b)\), substitute in \(\frac{dy}{dx} = \frac{x}{4y}\) to get the slope of the tangent line at that point. It's worth noting that the exact coordinates of the point, which might be given or estimated from the graph, are necessary to proceed with this step.

Key concepts

Explicit Function

An explicit function is one where the dependent variable, often denoted as \( y \), is expressed directly in terms of the independent variable \( x \). This is straightforward because you can see the relationship clearly laid out. For example, an explicit function could be written as \( y = f(x) \). In the exercise, we derived the explicit form by manipulating the implicit equation \( 4y^2 - x^2 = 7 \) to solve for \( y \).
This resulted in the function \( y = \sqrt{\frac{x^2 + 7}{4}} \). Explicit functions simplify finding derivatives because all variables are isolated, making calculations more straightforward and direct.

• Direct relationship between \( x \) and \( y \)
• Clear formula for calculation

Chain Rule

The chain rule is essential for differentiating composite functions, which are functions wrapped inside one another, such as \( y = g(f(x)) \). In simpler terms, it helps find the derivative of one function within another. The rule states \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \), where \( u \) is an intermediate variable substituting a function of \( x \).
In the exercise, when differentiating the explicit function \( y = \sqrt{(x^2 + 7) / 4} \), we applied the chain rule. First, compute the derivative of the outer function (the square root) and then multiply it by the derivative of the inner function (\( x^2 + 7 \)). This method ensures that all components of a complicated function are considered.

• Useful for nesting functions
• Breaks complex derivatives into manageable pieces

Tangent Line

A tangent line is a straight line that "just touches" a curve at a given point, without cutting through it. It's a geometric concept that illustrates how a curve behaves at exactly one point. The slope of this tangent line is the derivative at that point, providing a linear approximation of the curve's behavior.
In the given exercise, after finding the derivative \( \frac{dy}{dx} = \frac{x}{4y} \), you can use this derivative to determine how steep the tangent is at any specific point \((a, b)\). Just substitute \( a \) and \( b \) into the derivative to calculate the slope, which represents the tangent line at that point.

• Represents instantaneous rate of change
• Line that locally approximates curve

Derivative Estimation

Derivative estimation involves finding or guessing the slope of a curve at a point using a derivative expression. This is particularly useful when exact values are hard to compute or when you rely on visual graphs.
In the exercise, estimation is done by visually inspecting the graph alongside analytical methods. The derivative formula \( \frac{dy}{dx} \) allows us to make precise estimations by substituting known values of \( x \) and \( y \). Derivative estimation is a blend of graphical and analytical techniques that helps validate results from different approaches.

• Utilizes both graph and calculus
• Validates analytical solutions