Question Statement

Find the derivative of the function:

$y = \frac{2 - 3 x}{7 - x}$

with respect to $x$ .

Background and Explanation

This problem involves differentiating a rational function where both the numerator and denominator are linear expressions. Since the function is expressed as a quotient of two functions of $x$ , we must apply the quotient rule for differentiation rather than simple power rules.

Solution

We are given the function:

$y = \frac{2 - 3 x}{7 - x}$

To differentiate this rational function, we apply the quotient rule, which states that for a function $y = \frac{u}{v}$ :

$\frac{d y}{d x} = \frac{v \cdot \frac{d u}{d x} - u \cdot \frac{d v}{d x}}{v ^{2}}$

Step 1: Identify the numerator and denominator functions.

Let $u = 2 - 3 x$ , so $\frac{d u}{d x} = - 3$
Let $v = 7 - x$ , so $\frac{d v}{d x} = - 1$

Step 2: Apply the quotient rule formula.

$\frac{d y}{d x} = \frac{( 7 - x ) \frac{d}{d x} ( 2 - 3 x ) - ( 2 - 3 x ) \frac{d}{d x} ( 7 - x )}{( 7 - x ) ^{2}}$

Step 3: Compute the derivatives of the numerator and denominator.

$\frac{d}{d x} (2 - 3 x) = 0 - 3 = - 3$
$\frac{d}{d x} (7 - x) = 0 - 1 = - 1$

Substituting these back:

$\frac{d y}{d x} = \frac{( 7 - x ) \cdot ( - 3 ) - ( 2 - 3 x ) \cdot ( - 1 )}{( 7 - x ) ^{2}}$

Step 4: Expand and simplify the numerator.

$\frac{d y}{d x} = \frac{- 21 + 3 x - ( - 2 + 3 x )}{( 7 - x ) ^{2}}$

$\frac{d y}{d x} = \frac{- 21 + 3 x + 2 - 3 x}{( 7 - x ) ^{2}}$

Step 5: Combine like terms. Notice that the $+ 3 x$ and $- 3 x$ terms cancel each other out:

$\frac{d y}{d x} = \frac{- 19}{( 7 - x ) ^{2}}$

Thus, the derivative is:

$\frac{d y}{d x} = \frac{- 19}{( 7 - x ) ^{2}}$

Key Formulas or Methods Used

Quotient Rule: $\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}$
Power Rule for linear terms: $\frac{d}{d x} (a x + b) = a$
Algebraic simplification: Combining like terms ( $3 x - 3 x = 0$ )

Summary of Steps

Identify the numerator $u = 2 - 3 x$ and denominator $v = 7 - x$ of the rational function.
Differentiate both parts separately: $\frac{d u}{d x} = - 3$ and $\frac{d v}{d x} = - 1$ .
Apply the quotient rule formula: $\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$ .
Substitute the values: $\frac{( 7 - x ) ( - 3 ) - ( 2 - 3 x ) ( - 1 )}{( 7 - x ) ^{2}}$ .
Expand the numerator: $- 21 + 3 x + 2 - 3 x$ .
Simplify by combining like terms (the $x$ terms cancel) to get the final answer: $\frac{- 19}{( 7 - x ) ^{2}}$ .

Question Statement

Find the derivative of the function:

$y = \frac{2 - 3 x}{7 - x}$

with respect to $x$ .

Background and Explanation

Solution

We are given the function:

$y = \frac{2 - 3 x}{7 - x}$

To differentiate this rational function, we apply the quotient rule, which states that for a function $y = \frac{u}{v}$ :

$\frac{d y}{d x} = \frac{v \cdot \frac{d u}{d x} - u \cdot \frac{d v}{d x}}{v ^{2}}$

Step 1: Identify the numerator and denominator functions.

Let $u = 2 - 3 x$ , so $\frac{d u}{d x} = - 3$
Let $v = 7 - x$ , so $\frac{d v}{d x} = - 1$

Step 2: Apply the quotient rule formula.

$\frac{d y}{d x} = \frac{( 7 - x ) \frac{d}{d x} ( 2 - 3 x ) - ( 2 - 3 x ) \frac{d}{d x} ( 7 - x )}{( 7 - x ) ^{2}}$

Step 3: Compute the derivatives of the numerator and denominator.

$\frac{d}{d x} (2 - 3 x) = 0 - 3 = - 3$
$\frac{d}{d x} (7 - x) = 0 - 1 = - 1$

Substituting these back:

$\frac{d y}{d x} = \frac{( 7 - x ) \cdot ( - 3 ) - ( 2 - 3 x ) \cdot ( - 1 )}{( 7 - x ) ^{2}}$

Step 4: Expand and simplify the numerator.

$\frac{d y}{d x} = \frac{- 21 + 3 x - ( - 2 + 3 x )}{( 7 - x ) ^{2}}$

$\frac{d y}{d x} = \frac{- 21 + 3 x + 2 - 3 x}{( 7 - x ) ^{2}}$

Step 5: Combine like terms. Notice that the $+ 3 x$ and $- 3 x$ terms cancel each other out:

$\frac{d y}{d x} = \frac{- 19}{( 7 - x ) ^{2}}$

Thus, the derivative is:

$\frac{d y}{d x} = \frac{- 19}{( 7 - x ) ^{2}}$

Key Formulas or Methods Used

Quotient Rule: $\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}$
Power Rule for linear terms: $\frac{d}{d x} (a x + b) = a$
Algebraic simplification: Combining like terms ( $3 x - 3 x = 0$ )

Summary of Steps

Identify the numerator $u = 2 - 3 x$ and denominator $v = 7 - x$ of the rational function.
Differentiate both parts separately: $\frac{d u}{d x} = - 3$ and $\frac{d v}{d x} = - 1$ .
Apply the quotient rule formula: $\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$ .
Substitute the values: $\frac{( 7 - x ) ( - 3 ) - ( 2 - 3 x ) ( - 1 )}{( 7 - x ) ^{2}}$ .
Expand the numerator: $- 21 + 3 x + 2 - 3 x$ .
Simplify by combining like terms (the $x$ terms cancel) to get the final answer: $\frac{- 19}{( 7 - x ) ^{2}}$ .

2.5 Q-8

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.5 Q-8

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps