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. $$\sqrt[3]{x}+\sqrt[3]{y^{4}}=2 ;(1,1)$$

Short answer

Answer: The slope of the curve at the point (1,1) is \(-\frac{3}{4}\).

Step-by-step solution

Implicit Differentiation

To find the derivative \(\frac{dy}{dx}\), differentiate both sides of the equation with respect to \(x\): $$\frac{d(\sqrt[3]{x})}{dx}+\frac{d(\sqrt[3]{y^4})}{dx}= \frac{d(2)}{dx}$$ Now let's find the derivatives for both terms: $$\frac{d(\sqrt[3]{x})}{dx}=\frac{1}{3x^{2/3}}$$ and $$\frac{d(\sqrt[3]{y^4})}{dx}=\frac{d(\sqrt[3]{y^4})}{dy}\cdot\frac{dy}{dx}=\frac{4}{3}y^{1/3}\cdot\frac{dy}{dx}$$ Now, substitute the derivatives back to the equation: $$\frac{1}{3x^{2/3}}+\frac{4}{3}y^{1/3}\cdot\frac{dy}{dx}=0$$ Now, find an expression for \(\frac{dy}{dx}\) by solving: $$\frac{dy}{dx}=-\frac{1}{4y^{1/3}}\cdot3x^{2/3}$$ $$\frac{dy}{dx}=-\frac{3x^{2/3}}{4y^{1/3}}$$

Find the slope at the given point (1,1)

Now, we will calculate the slope of the curve at the point (1,1) by substituting the x and y values in the derivative expression we found in Step 1: $$\frac{dy}{dx}=-\frac{3(1)^{2/3}}{4(1)^{1/3}}$$ $$\frac{dy}{dx}=-\frac{3}{4}$$ So, the slope of the curve at the point (1,1) is $$-\frac{3}{4}$$.

Key concepts

Derivative

Implicit differentiation is a technique used to find the derivative of functions that are not isolated, meaning they are mixed together in an equation. In the given problem, we are dealing with a function involving both \(x\) and \(y\):

• \(\sqrt[3]{x} + \sqrt[3]{y^4} = 2\)

To find the derivative \(\frac{dy}{dx}\), we differentiate both sides of the equation with respect to \(x\) using implicit differentiation. This involves:

• Applying the chain rule for each term.
• Recognizing that when differentiating \(y\), we must multiply by \(\frac{dy}{dx}\).

This results in a linear equation where you solve for \(\frac{dy}{dx}\). It helps you understand how \(y\) changes with \(x\), even when they are interlinked.

Slope

The slope of a curve at a particular point is essentially the gradient or steepness of the tangent line to the curve at that point. It tells us how sharply the curve is rising or falling:

• For a positive slope, the curve ascends.
• For a negative slope, the curve descends.
• A zero slope indicates a flat, horizontal tangent.

In our problem, once we found the derivative \(\frac{dy}{dx}\), the value of this derivative at a specific point gives us the slope at that point. For algebraic curves, the slope can vary across different points on the curve, reflecting the curve's changing direction and steepness.

Point (1,1)

When dealing with functions and their derivatives, knowing the specific point at which to evaluate the derivative is crucial. For this problem:

• The given point is \((1,1)\).
• You use these coordinates to substitute \(x=1\) and \(y=1\) into the derivative expression \(\frac{dy}{dx}\).

The coordinates of the point provide specific numerical values for \(x\) and \(y\). This allows us to calculate the exact slope of the curve at that particular point, showing precisely how steep the curve is at \((1,1)\). Without this point, you wouldn't get the precise slope value needed for detailed analysis.

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

In this exercise, finding an expression for \(\frac{dy}{dx}\) is key to understanding the relationship between \(y\) and \(x\) implicitly defined in the original equation. The step-by-step process is outlined as follows:

• Differentiate each term separately with respect to \(x\).
• Find the derivative of the constant, which is zero, leaving the other terms to solve for \(\frac{dy}{dx}\).

The expression we derived:

• \(\frac{dy}{dx} = -\frac{3x^{2/3}}{4y^{1/3}}\)

shows how \(y\) changes with respect to \(x\) for any point on the curve. This expression lets us plug in any values of \(x\) and \(y\) to find the slope at any point, making it an incredibly powerful tool for understanding the curve's geometry.