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In calculus, the quotient rule is a method for finding the derivative of a quotient of two functions. When we have a function expressed as a fraction, such as \( y = \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \), the quotient rule provides a systematic way to find its derivative. Given by:\[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\]**Steps to Apply the Quotient Rule:**

• Identify the numerator function \( u \) and the denominator function \( v \).
• Calculate the derivative of \( u \), denoted as \( u' \).
• Calculate the derivative of \( v \), denoted as \( v' \).
• Substitute these values into the quotient rule formula.
• Simplify the expression to get the final derivative.

In the given exercise, \( u = x+3 \) and \( v = 1-x \). We find \( u' = 1 \) and \( v' = -1 \) and then apply the quotient rule to arrive at the derivative:\[y' = \frac{4}{(1-x)^2}\] This process highlights how the quotient rule breaks down complex derivatives into manageable steps.

The tangent line is a straight line that touches a curve at a single point without crossing it at that point. This line represents the "instantaneous rate of change" of the function at a given point, which is the same as the derivative's value at that point. **Understanding Tangent Lines:**

• At any given point \( x \) on the curve, the slope of the tangent line is equal to the derivative of the function at that point, \( y'(x) \).
• The equation of the tangent line can be derived using the slope-point form: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency.

In the exercise, after finding the derivative \( y' = \frac{4}{(1-x)^2} \), substituting \( x = -2 \) gives us the slope of the tangent line:\[y'(-2) = \frac{4}{9}\]This implies that at \( x = -2 \), the curve is rising at a rate of \( \frac{4}{9} \), providing insight into how steep or gentle the curve is at that point.

Calculating derivatives is a fundamental skill in calculus. It lets us understand how a function is changing at any specific point. The derivative of a function gives the slope of the tangent line to the graph of the function at that point.**Steps for Derivative Calculation:**

• Identify the function you are working with. It might already be in a simplifiable form, or you might need to rewrite it using algebraic manipulations.
• Apply the appropriate rules, such as product, sum, or quotient rules, to find the derivative.
• Simplify the derivative expression to make further calculations easier.
• Substitute any given values into the derivative to find specific information like slopes or rates of change.

For the function \( y = \frac{x+3}{1-x} \), we recognize it as a quotient, suggesting direct application of the quotient rule. After simplifying, the derivative calculated is \( y' = \frac{4}{(1-x)^2} \). When substituting \( x = -2 \), the derivative helps us determine the rate at which the function is changing at that precise point, resulting in a slope of \( \frac{4}{9} \) for the tangent line.