Question Statement

Find the derivative of the function:

$y = \frac{( x ^{2} + 1 ) ^{2}}{3 x - 2}$

with respect to $x$ .

Background and Explanation

This problem involves differentiating a quotient where the numerator is a composite function (a polynomial raised to a power). You will need to apply the quotient rule for the overall division structure, combined with the chain rule to differentiate the squared term in the numerator.

Solution

We are given the function:

$y = \frac{( x ^{2} + 1 ) ^{2}}{3 x - 2}$

To differentiate this, we apply the quotient rule:

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

Let $u = (x^{2} + 1)^{2}$ and $v = 3 x - 2$ .

Step 1: Find $\frac{d u}{d x}$ using the chain rule. $\frac{d}{d x} (x^{2} + 1)^{2} = 2 (x^{2} + 1)^{2 - 1} \cdot \frac{d}{d x} (x^{2} + 1) = 2 (x^{2} + 1) \cdot 2 x = 4 x (x^{2} + 1)$

Step 2: Find $\frac{d v}{d x}$ . $\frac{d}{d x} (3 x - 2) = 3$

Step 3: Apply the quotient rule formula. $\frac{d y}{d x} = \frac{( 3 x - 2 ) \cdot 4 x ( x ^{2} + 1 ) - ( x ^{2} + 1 ) ^{2} \cdot 3}{( 3 x - 2 ) ^{2}}$

Step 4: Factor out common terms in the numerator. Notice that $(x^{2} + 1)$ is common to both terms in the numerator: $\frac{d y}{d x} = \frac{( x ^{2} + 1 ) [ 4 x ( 3 x - 2 ) - 3 ( x ^{2} + 1 ) ]}{( 3 x - 2 ) ^{2}}$

Step 5: Simplify the expression inside the brackets. $4 x (3 x - 2) - 3 (x^{2} + 1) = 12 x^{2} - 8 x - 3 x^{2} - 3 = 9 x^{2} - 8 x - 3$

So we have: $\frac{d y}{d x} = \frac{( x ^{2} + 1 ) ( 9 x ^{2} - 8 x - 3 )}{( 3 x - 2 ) ^{2}}$

Step 6: Expand the numerator (optional, for fully expanded form). $(x^{2} + 1) (9 x^{2} - 8 x - 3) = 9 x^{4} - 8 x^{3} - 3 x^{2} + 9 x^{2} - 8 x - 3 = 9 x^{4} - 8 x^{3} + 6 x^{2} - 8 x - 3$

Therefore, the derivative is:

$\frac{d y}{d x} = \frac{( x ^{2} + 1 ) ( 9 x ^{2} - 8 x - 3 )}{( 3 x - 2 ) ^{2}}$

or in expanded form:

$\frac{d y}{d x} = \frac{9 x ^{4} - 8 x ^{3} + 6 x ^{2} - 8 x - 3}{( 3 x - 2 ) ^{2}}$

Key Formulas or Methods Used

Quotient Rule: $\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}$
Chain Rule: $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$
Power Rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$

Summary of Steps

Identify components: Set $u = (x^{2} + 1)^{2}$ and $v = 3 x - 2$
Differentiate $u$ : Apply chain rule to get $u^{'} = 4 x (x^{2} + 1)$
Differentiate $v$ : Apply power rule to get $v^{'} = 3$
Apply quotient rule: Substitute into $\frac{v u ^{'} - u v ^{'}}{v ^{2}}$
Factor numerator: Extract the common factor $(x^{2} + 1)$
Simplify: Expand and combine like terms inside the brackets
Final form: Present as factored or fully expanded fraction

Question Statement

Find the derivative of the function:

$y = \frac{( x ^{2} + 1 ) ^{2}}{3 x - 2}$

with respect to $x$ .

Background and Explanation

Solution

We are given the function:

$y = \frac{( x ^{2} + 1 ) ^{2}}{3 x - 2}$

To differentiate this, we apply the quotient rule:

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

Let $u = (x^{2} + 1)^{2}$ and $v = 3 x - 2$ .

Step 1: Find $\frac{d u}{d x}$ using the chain rule. $\frac{d}{d x} (x^{2} + 1)^{2} = 2 (x^{2} + 1)^{2 - 1} \cdot \frac{d}{d x} (x^{2} + 1) = 2 (x^{2} + 1) \cdot 2 x = 4 x (x^{2} + 1)$

Step 2: Find $\frac{d v}{d x}$ . $\frac{d}{d x} (3 x - 2) = 3$

Step 3: Apply the quotient rule formula. $\frac{d y}{d x} = \frac{( 3 x - 2 ) \cdot 4 x ( x ^{2} + 1 ) - ( x ^{2} + 1 ) ^{2} \cdot 3}{( 3 x - 2 ) ^{2}}$

Step 5: Simplify the expression inside the brackets. $4 x (3 x - 2) - 3 (x^{2} + 1) = 12 x^{2} - 8 x - 3 x^{2} - 3 = 9 x^{2} - 8 x - 3$

So we have: $\frac{d y}{d x} = \frac{( x ^{2} + 1 ) ( 9 x ^{2} - 8 x - 3 )}{( 3 x - 2 ) ^{2}}$

Step 6: Expand the numerator (optional, for fully expanded form). $(x^{2} + 1) (9 x^{2} - 8 x - 3) = 9 x^{4} - 8 x^{3} - 3 x^{2} + 9 x^{2} - 8 x - 3 = 9 x^{4} - 8 x^{3} + 6 x^{2} - 8 x - 3$

Therefore, the derivative is:

$\frac{d y}{d x} = \frac{( x ^{2} + 1 ) ( 9 x ^{2} - 8 x - 3 )}{( 3 x - 2 ) ^{2}}$

or in expanded form:

$\frac{d y}{d x} = \frac{9 x ^{4} - 8 x ^{3} + 6 x ^{2} - 8 x - 3}{( 3 x - 2 ) ^{2}}$

Key Formulas or Methods Used

Quotient Rule: $\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}$
Chain Rule: $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$
Power Rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$

Summary of Steps

Identify components: Set $u = (x^{2} + 1)^{2}$ and $v = 3 x - 2$
Differentiate $u$ : Apply chain rule to get $u^{'} = 4 x (x^{2} + 1)$
Differentiate $v$ : Apply power rule to get $v^{'} = 3$
Apply quotient rule: Substitute into $\frac{v u ^{'} - u v ^{'}}{v ^{2}}$
Factor numerator: Extract the common factor $(x^{2} + 1)$
Simplify: Expand and combine like terms inside the brackets
Final form: Present as factored or fully expanded fraction

2.5 Q-12

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.5 Q-12

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps