91Ó°ÊÓ

Implicit differentiation is a technique used to differentiate equations that define a function implicitly rather than explicitly. In the given exercise, we are tasked to differentiate the equation \( x^3 = y^2 (2 - x) \) implicitly to find the slope of the tangent line at a specific point.
To begin, we differentiate both sides of the equation with respect to \( x \), treating \( y \) as a function of \( x \). This means we apply the chain rule to terms that involve \( y \).
Implicit differentiation is especially useful when dealing with equations that are difficult or impossible to solve for \( y \) explicitly.

In summary:

• Differentiating implicitly allows us to work directly with functions given in implicit form.
• We treat \( y \) as a function of \( x \) and apply the chain rule accordingly.

Implicit differentiation becomes essential in finding the derivative needed for our tangent line calculation.

The product rule is a method used to differentiate products of two or more functions. In the exercise, the product rule helps us differentiate the right-hand side of the implicit equation \[ x^3 = y^2 (2 - x) \].
The product rule states that if we have a function \( u(x) \) and \( v(x) \), then the derivative of their product is given by: \[ \frac{d}{dx} [u(x) \times v(x)] = u'(x) \times v(x) + u(x) \times v'(x) \].

Applying the product rule to \( y^2 (2 - x) \), we get:
\[ \frac{d}{dx} [y^2 (2 - x)] = \frac{d}{dx} [y^2] \times (2 - x) + y^2 \times \frac{d}{dx} [2 - x] \].
This lets us handle more complex differentiation.

In summary:

• The product rule assists us when differentiating the product of two functions.
• It breaks down the differentiation process into simpler parts.

This improves accuracy and maintains structure during differentiation.

The chain rule is a formula for computing the derivative of the composition of two or more functions. In this exercise, it is essential for differentiating the term \( y^2 \).
The chain rule states: If a variable \( z \) depends on the variable \( y \), and \( y \) depends on \( x \), then \[ \frac{dz}{dx} = \frac{dz}{dy} \times \frac{dy}{dx} \].
When differentiating \( y^2 \) with respect to \( x \), where \( y \) is a function of \( x \), we get:
\[ \frac{d}{dx} [y^2] = 2y \times \frac{dy}{dx} \]. This encapsulates the essence of the chain rule, recognizing that \( y \) itself needs differentiation.

In summary:

• The chain rule simplifies differentiation of composite functions.
• It helps us understand how changes in one variable propagate through others.

Using the chain rule allows us to successfully differentiate implicit functions like those in this exercise.

A derivative represents the rate at which a function is changing at any given point. In the context of the tangent line problem, we need the derivative to determine the slope.
After differentiating both sides of the equation \[ x^3 = y^2 (2 - x) \], we combine the results to isolate \( \frac{dy}{dx} \):
\[ 3x^2 = 2y \frac{dy}{dx} (2 - x) + y^2 (-1) \] simplifies to:
\[ \frac{dy}{dx} = \frac{3x^2 + y^2}{2y(2 - x)} \].
With the derivative in hand, we can substitute the given point \( (-0.8, 0.384) \) to find the slope at that point, resulting in about \( 0.9616 \).

In summary:

• Derivatives determine how a function's output changes with respect to its input.
• In tangent line calculations, they provide the necessary slope at a specific point.

Understanding the concept of a derivative ensures better comprehension of the tangent line equation.

The point-slope form of a line equation is used to write the equation given a point and a slope. This form is crucial in forming the equation of the tangent line.
Once we have the slope \( m \) at the given point \( (-0.8, 0.384) \), the point-slope form is expressed as:
\[ y - y_1 = m(x - x_1) \].
Substituting \( m = 0.9616 \), \( x_1 = -0.8 \), and \( y_1 = 0.384 \), we get:
\[ y - 0.384 = 0.9616 (x + 0.8) \]. Simplifying, the equation becomes:
\[ y = 0.9616x + 1.15328 \].

In summary:

• The point-slope form connects the slope of a line to a point on the line.
• It makes it easy to derive the tangent line equation using known values.

Point-slope form solidifies our understanding of how to write linear equations for specific scenarios like tangent lines.