Question

Carry out the following steps. \(x.\) a. Use implicit differentiation to find \(\frac{d y}{d x}.\) b. Find the slope of the curve at the given point. $$x y^{5 / 2}+x^{3 / 2} y=12 ;(4,1)$$

Short answer

Answer: The slope of the curve at the point (4,1) is \(-\frac{2}{9}\).

Step-by-step solution

Use implicit differentiation to find \(\frac{dy}{dx}\)

The given implicit function can be written as \(x y^{5/2} + x^{3/2} y = 12\). We will differentiate both sides of the equation with respect to x. The derivative of \(x y^{5/2}\) with respect to x is \(\frac{d(x y^{5/2})}{dx}\) and the derivative of \(x^{3/2} y\) with respect to x is \(\frac{d(x^{3/2} y)}{dx}\). The derivative of 12 will be 0. Using implicit differentiation, \(\frac{d(x y^{5/2})}{dx} + \frac{d(x^{3/2} y)}{dx} = 0\)

Differentiate the terms and solve for \(\frac{dy}{dx}\)

First, differentiate \(x y^{5/2}\): Product rule: \(\frac{d(uv)}{dx}= u \frac{dv}{dx} + v \frac{du}{dx}\), where \(u = x\) and \(v = y^{5/2}\). Therefore, \(\frac{d(x y^{5/2})}{dx} = x \frac{d(y^{5/2})}{dx} + y^{5/2} \frac{dx}{dx}\) \(\implies \frac{d(x y^{5/2})}{dx} = x (5/2) y^{3/2} (\frac{dy}{dx}) + y^{5/2}\) Now, differentiate \(x^{3/2} y\): Here, \(u = x^{3/2}\) and \(v = y\). \(\frac{d(x^{3/2} y)}{dx} = x^{3/2} \frac{dy}{dx} + y (\frac{3}{2}) x^{1/2}\) So, our implicit differentiation equation becomes: \(x (5/2) y^{3/2} (\frac{dy}{dx}) + y^{5/2} + x^{3/2} \frac{dy}{dx} + y (\frac{3}{2}) x^{1/2} = 0\) For simplification: \((5/2) x y^{3/2} \frac{dy}{dx} + y^{5/2} + x^{3/2} \frac{dy}{dx} + \frac{3}{2} x^{1/2} y = 0\) Now, let's solve for \(\frac{dy}{dx}\): \(\frac{dy}{dx} (\frac{5}{2} x y^{3/2} + x^{3/2}) = -y^{5/2} - \frac{3}{2} x^{1/2} y\) Therefore, \(\frac{dy}{dx} = \frac{-y^{5/2} - \frac{3}{2} x^{1/2} y}{\frac{5}{2} x y^{3/2} + x^{3/2}}\)

Find the slope of the curve at the given point (4,1)

To find the slope of the curve at the point (4,1), plug x = 4 and y = 1 into the expression we found for \(\frac{dy}{dx}\): \(\frac{dy}{dx} |_{(4,1)} = \frac{-(1^{5/2}) - \frac{3}{2} (4^{1/2})(1)}{\frac{5}{2} (4)(1^{3/2}) + (4^{3/2})}\) \( \implies \frac{dy}{dx} |_{(4,1)} = \frac{-1 - 3}{10 + 8} = \frac{-4}{18} = -\frac{2}{9}\) The slope of the curve at the point (4,1) is \(-\frac{2}{9}\).

Key concepts

Product Rule

The product rule is a crucial tool in calculus that helps differentiate expressions where two functions are multiplied together. When you have a function that is the product of two simpler functions, the product rule provides a systematic way to find its derivative.
To use the product rule, suppose you have two functions, \(u\) and \(v\). If you want to find the derivative of their product \(uv\), the product rule states:

• \(\frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}\)

This means you differentiate the first function \(u\) and multiply it by the second function \(v\) as is, and vice versa. When applying this rule, pay attention to each term, ensuring you properly apply derivatives to each part.
In our exercise, we have expressions like \(x y^{5/2}\) where applying the product rule helps to successfully differentiate both parts with respect to \(x\). Find each component's derivative and ensure that they are correctly combined.

Derivative

The derivative is fundamental in calculus, representing the rate at which a function changes. It provides a way to determine slopes of tangent lines to curves, essential in understanding the behavior of functions.
When we talk about derivatives, especially in implicit differentiation, it is crucial to keep in mind that you often differentiate terms with respect to \(x\) even if they also involve \(y\). This requires careful application of rules like the product rule and chain rule. For example, the term \(y^{5/2}\) introduces an implicit relationship with \(x\), necessitating careful handling.
In our step-by-step solution, derivatives help determine \(\frac{dy}{dx}\) despite \(y\) implicitly depending on \(x\). Each derivative step brings insights into how every term relates to the changes in \(x\). Thus, derivatives enable us to uncover these implicit relationships within the equation.

Slope of the Curve

The slope of a curve at a particular point provides the steepness and direction of the curve. In the context of implicit differentiation, finding \(\frac{dy}{dx}\) is pivotal as it represents the slope of the curve in Cartesian coordinates.
The question involves determining the slope using implicit differentiation at the point (4,1). Essentially, we substitute the values of \(x\) and \(y\) back into our expression from which we've already derived \(\frac{dy}{dx}\). By computing this value, we measure how steeply the curve rises or falls at that specific point.
This slope can also reveal whether the curve is increasing, decreasing, or remaining constant at the given point, hence its practical importance in sketching and analyzing the behavior of curves.

Implicit Function

Implicit functions are those where \(y\) is not expressed explicitly in terms of \(x\). Instead, \(y\) and \(x\) are intertwined within an equation, making direct solving for \(y\) challenging. This is where implicit differentiation comes into play.
When dealing with implicit functions, we differentiate both sides of the equation with respect to \(x\) while treating \(y\) as a function of \(x\). Such functions are an integral part of mathematical modeling, as they can represent complex relationships between variables.
In our exercise, the given function \(x y^{5/2} + x^{3/2} y = 12\) is implicit. By applying implicit differentiation, you efficiently solve for \(\frac{dy}{dx}\) without needing to isolate \(y\) as an explicit function of \(x\). This technique simplifies the process of understanding how variables interrelate within an implicit context.