Question Statement

Find $\frac{d y}{d x}$ for the function:

$y = \frac{3 x ^{2} + 5}{3 x - 1}$

Background and Explanation

This problem requires differentiating a rational function using the quotient rule, which applies when a function is expressed as the ratio of two differentiable functions. You'll also need the power rule for differentiating polynomial terms.

Solution

We are given the function:

$y = \frac{3 x ^{2} + 5}{3 x - 1}$

To differentiate this with respect to $x$ , we apply the quotient rule. Let $u = 3 x^{2} + 5$ and $v = 3 x - 1$ .

First, find the derivatives of the numerator and denominator:

$\frac{d}{d x} (3 x^{2} + 5) = 6 x$
$\frac{d}{d x} (3 x - 1) = 3$

Now apply the quotient rule formula $\frac{d y}{d x} = \frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$ :

$\frac{d y}{d x} = \frac{( 3 x - 1 ) \cdot \frac{d}{d x} ( 3 x ^{2} + 5 ) - ( 3 x ^{2} + 5 ) \cdot \frac{d}{d x} ( 3 x - 1 )}{( 3 x - 1 ) ^{2}} = \frac{( 3 x - 1 ) \cdot ( 6 x ) - ( 3 x ^{2} + 5 ) \cdot ( 3 )}{( 3 x - 1 ) ^{2}} = \frac{18 x ^{2} - 6 x - 9 x ^{2} - 15}{( 3 x - 1 ) ^{2}} = \frac{9 x ^{2} - 6 x - 15}{( 3 x - 1 ) ^{2}} = \frac{3 ( 3 x ^{2} - 2 x - 5 )}{( 3 x - 1 ) ^{2}}$

Explanation of steps:

Line 1: Apply the quotient rule structure: $\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$
Line 2: Compute the derivatives: $(3 x^{2} + 5)^{'} = 6 x$ and $(3 x - 1)^{'} = 3$
Line 3: Expand the numerator: $(3 x - 1) (6 x) = 18 x^{2} - 6 x$ and $(3 x^{2} + 5) (3) = 9 x^{2} + 15$ , then distribute the negative sign
Line 4: Combine like terms in the numerator: $18 x^{2} - 9 x^{2} = 9 x^{2}$ , leaving $9 x^{2} - 6 x - 15$
Line 5: Factor out the common factor of $3$ from the numerator to obtain the simplified form

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: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ (used to differentiate $3 x^{2}$ and $3 x$ )
Constant Rule: $\frac{d}{d x} (c) = 0$ (derivative of constants $5$ and $- 1$ is zero)

Summary of Steps

Identify components: Let $u = 3 x^{2} + 5$ (numerator) and $v = 3 x - 1$ (denominator)
Differentiate separately: Find $u^{'} = 6 x$ and $v^{'} = 3$
Apply quotient rule: Substitute into $\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$
Expand and simplify: Multiply out the terms in the numerator and combine like terms
Factor: Extract the common factor of $3$ from the numerator for the final simplified form

Solution

We are given the function:

y = \frac{3 x ^{2} + 5}{3 x - 1}

To differentiate this with respect to

x

, we apply the quotient rule. Let

u = 3 x^{2} + 5

and

v = 3 x - 1

First, find the derivatives of the numerator and denominator:

\frac{d}{d x} (3 x^{2} + 5) = 6 x

\frac{d}{d x} (3 x - 1) = 3

Now apply the quotient rule formula

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

\frac{d y}{d x} = \frac{( 3 x - 1 ) \cdot \frac{d}{d x} ( 3 x ^{2} + 5 ) - ( 3 x ^{2} + 5 ) \cdot \frac{d}{d x} ( 3 x - 1 )}{( 3 x - 1 ) ^{2}} = \frac{( 3 x - 1 ) \cdot ( 6 x ) - ( 3 x ^{2} + 5 ) \cdot ( 3 )}{( 3 x - 1 ) ^{2}} = \frac{18 x ^{2} - 6 x - 9 x ^{2} - 15}{( 3 x - 1 ) ^{2}} = \frac{9 x ^{2} - 6 x - 15}{( 3 x - 1 ) ^{2}} = \frac{3 ( 3 x ^{2} - 2 x - 5 )}{( 3 x - 1 ) ^{2}}

Explanation of steps:

Line 1: Apply the quotient rule structure:

\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}

Line 2: Compute the derivatives:

(3 x^{2} + 5)^{'} = 6 x

and

(3 x - 1)^{'} = 3

Line 3: Expand the numerator:

(3 x - 1) (6 x) = 18 x^{2} - 6 x

and

(3 x^{2} + 5) (3) = 9 x^{2} + 15

, then distribute the negative sign

Line 4: Combine like terms in the numerator:

18 x^{2} - 9 x^{2} = 9 x^{2}

, leaving

9 x^{2} - 6 x - 15

Line 5: Factor out the common factor of

3

from the numerator to obtain the simplified form

Summary of Steps

Identify components: Let

u = 3 x^{2} + 5

(numerator) and

v = 3 x - 1

(denominator)

Differentiate separately: Find

u^{'} = 6 x

and

v^{'} = 3

Apply quotient rule: Substitute into

\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}

Expand and simplify: Multiply out the terms in the numerator and combine like terms

Factor: Extract the common factor of

3

from the numerator for the final simplified form

2.5 Q-6

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.5 Q-6

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps