Question

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

Short answer

Answer: The slope of the curve at the point (1, 1) is -1.

Step-by-step solution

Implicit differentiation with respect to x

To differentiate the equation \(x^{\frac{2}{3}} + y^{\frac{2}{3}} = 2\) implicitly, we will differentiate both sides of the equation with respect to \(x\): $$\frac{d(x^{\frac{2}{3}})}{dx} + \frac{d(y^{\frac{2}{3}})}{dx} = \frac{d(2)}{dx}$$

Solve for \(\frac{dy}{dx}\)

Now, we need to differentiate both terms independently: $$\frac{d(x^{\frac{2}{3}})}{dx} = \frac{2}{3}x^{\frac{-1}{3}}$$ Since we need to differentiate \(y^{\frac{2}{3}}\) with respect to \(x\), not \(y\), we will use the chain rule: $$\frac{d(y^{\frac{2}{3}})}{dx} = \frac{2}{3}y^{\frac{-1}{3}}\frac{dy}{dx}$$ The derivative of the constant term \(2\) is \(0\). By substituting the derivatives into the original equation, we get: $$\frac{2}{3}x^{\frac{-1}{3}} + \frac{2}{3}y^{\frac{-1}{3}}\frac{dy}{dx} = 0$$ Now, let's solve for \(\frac{dy}{dx}\): $$\frac{dy}{dx} = -\frac{x^{\frac{-1}{3}}}{y^{\frac{-1}{3}}}$$

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

Now we will find the slope of the curve at the given point (1, 1) by substituting the coordinates into the equation for \(\frac{dy}{dx}\): $$\frac{dy}{dx} = -\frac{1^{\frac{-1}{3}}}{1^{\frac{-1}{3}}} = -1$$ So, the slope of the curve at the point (1, 1) is -1.

Key concepts

Slope of a Curve

Understanding the slope of a curve is key when studying calculus. The slope tells you how steep a curve is at a particular point, representing the rate at which one variable changes with respect to another. In a more familiar setting, think of the slope as the incline you experience when you walk up or down a hill. For mathematical curves, we often want to find the slope at specific points. This allows us to understand how the curve behaves locally. For example, given the curve defined by the equation \(x^{2/3} + y^{2/3} = 2\), finding the slope at the point (1,1) helps illustrate the curve's behavior at this precise location.

• To find this slope, we first need the derivative, \(\frac{dy}{dx}\), meaning the change in \(y\) with respect to \(x\).
• This derivative can be thought of as the slope of the tangent line to the curve at the given point.

Whenever you find \(\frac{dy}{dx}\) for a curve, you are essentially calculating this slope.

Chain Rule

The chain rule is a fundamental tool in calculus, especially when dealing with composite functions. It allows us to find the derivative of functions that are nested within each other. When we differentiate implicitly, like in our exercise, the chain rule becomes very handy. Imagine you want to differentiate \(y^{2/3}\) with respect to \(x\). Normally, we'd differentiate with respect to \(y\) since \(y\) is the base, but we need it in terms of \(x\). Here's where the chain rule comes in:

• First, differentiate \(y^{2/3}\) with respect to \(y\), giving us \(\frac{2}{3}y^{-1/3}\).
• Then, multiply by \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\), resulting in \(\frac{2}{3}y^{-1/3}\cdot\frac{dy}{dx}\).

This multiplication step is crucial and captures how the function \(y\) itself changes in relation to \(x\). The chain rule keeps everything linked, ensuring the right transformations are applied when differentiating composite layers.

Derivative Calculations

Derivative calculations involve finding the rate of change of one variable with respect to another. In calculus, this process is vital for understanding and predicting how functions behave. Let’s break down how we find the derivative in the given exercise:1. **Implicit Differentiation**: This approach is used when differentiating equations not solved for one variable in terms of another, like our equation \(x^{2/3} + y^{2/3} = 2\).

• We differentiated both sides concerning \(x\), getting \(\frac{d(x^{2/3})}{dx} + \frac{d(y^{2/3})}{dx} = 0\).

2. **Solve for \(\frac{dy}{dx}\)**: After applying the derivatives, we solve for our target derivative:

• First component gives \(\frac{2}{3}x^{-1/3}\).
• Second component, using chain rule, gives \(\frac{2}{3}y^{-1/3}\cdot\frac{dy}{dx}\).
• Plug into the equation and solve: \(\frac{dy}{dx} = -\frac{x^{-1/3}}{y^{-1/3}}\).

This gives us a formula for the slope at any point \((x,y)\). Calculating these derivatives accurately is crucial for extracting meaningful information from mathematical relationships.