Question Statement

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

$y = \frac{s i n x}{x ^{2} + s i n x}$

Background and Explanation

This problem requires differentiating a rational function where both the numerator and denominator contain trigonometric and polynomial terms. You will need to apply the quotient rule for differentiation, along with basic derivative formulas for sine functions and power rules.

Solution

We are given the function:

$y = \frac{s i n x}{x ^{2} + s i n x}$

To differentiate this with respect to $x$ , we apply the quotient rule:

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

Step 1: Identify the numerator and denominator functions.

Let $u = sin x$ , so $\frac{d u}{d x} = cos x$
Let $v = x^{2} + sin x$ , so $\frac{d v}{d x} = 2 x + cos x$

Step 2: Substitute into the quotient rule formula.

$\frac{d y}{d x} = \frac{( x ^{2} + s i n x ) \cdot \frac{d}{d x} ( s i n x ) - s i n x \cdot \frac{d}{d x} ( x ^{2} + s i n x )}{( x ^{2} + s i n x ) ^{2}}$

Step 3: Compute the derivatives in the numerator.

$\frac{d y}{d x} = \frac{( x ^{2} + s i n x ) \cdot c o s x - s i n x \cdot ( 2 x + c o s x )}{( x ^{2} + s i n x ) ^{2}}$

Step 4: Expand the numerator by distributing the terms.

$\frac{d y}{d x} = \frac{x ^{2} c o s x + s i n x c o s x - 2 x s i n x - s i n x c o s x}{( x ^{2} + s i n x ) ^{2}}$

Step 5: Simplify by combining like terms. Notice that $+ sin x cos x$ and $- sin x cos x$ cancel each other out.

$\frac{d y}{d x} = \frac{x ^{2} c o s x - 2 x s i n x}{( x ^{2} + s i n x ) ^{2}}$

Final Answer:

$\frac{d y}{d x} = \frac{x ^{2} c o s x - 2 x s i n x}{( x ^{2} + s i n x ) ^{2}}$

Key Formulas or Methods Used

Quotient Rule: $\frac{d}{d x} (\frac{u}{v}) = \frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$
Derivative of sine: $\frac{d}{d x} (sin x) = cos x$
Power Rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ (used for the $x^{2}$ term)
Sum Rule: $\frac{d}{d x} (f (x) + g (x)) = f^{'} (x) + g^{'} (x)$

Summary of Steps

Identify components: Recognize $u = sin x$ and $v = x^{2} + sin x$ for the quotient rule
Differentiate separately: Find $u^{'} = cos x$ and $v^{'} = 2 x + cos x$
Apply quotient rule: Substitute into $\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$
Expand numerator: Distribute to get $x^{2} cos x + sin x cos x - 2 x sin x - sin x cos x$
Simplify: Cancel the $sin x cos x$ terms to obtain the final result $\frac{x ^{2} c o s x - 2 x s i n x}{( x ^{2} + s i n x ) ^{2}}$

Question Statement

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

$y = \frac{s i n x}{x ^{2} + s i n x}$

Background and Explanation

Solution

We are given the function:

$y = \frac{s i n x}{x ^{2} + s i n x}$

To differentiate this with respect to $x$ , we apply the quotient rule:

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

Step 1: Identify the numerator and denominator functions.

Let $u = sin x$ , so $\frac{d u}{d x} = cos x$
Let $v = x^{2} + sin x$ , so $\frac{d v}{d x} = 2 x + cos x$

Step 2: Substitute into the quotient rule formula.

$\frac{d y}{d x} = \frac{( x ^{2} + s i n x ) \cdot \frac{d}{d x} ( s i n x ) - s i n x \cdot \frac{d}{d x} ( x ^{2} + s i n x )}{( x ^{2} + s i n x ) ^{2}}$

Step 3: Compute the derivatives in the numerator.

$\frac{d y}{d x} = \frac{( x ^{2} + s i n x ) \cdot c o s x - s i n x \cdot ( 2 x + c o s x )}{( x ^{2} + s i n x ) ^{2}}$

Step 4: Expand the numerator by distributing the terms.

$\frac{d y}{d x} = \frac{x ^{2} c o s x + s i n x c o s x - 2 x s i n x - s i n x c o s x}{( x ^{2} + s i n x ) ^{2}}$

Step 5: Simplify by combining like terms. Notice that $+ sin x cos x$ and $- sin x cos x$ cancel each other out.

$\frac{d y}{d x} = \frac{x ^{2} c o s x - 2 x s i n x}{( x ^{2} + s i n x ) ^{2}}$

Final Answer:

$\frac{d y}{d x} = \frac{x ^{2} c o s x - 2 x s i n x}{( x ^{2} + s i n x ) ^{2}}$

Key Formulas or Methods Used

Quotient Rule: $\frac{d}{d x} (\frac{u}{v}) = \frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$
Derivative of sine: $\frac{d}{d x} (sin x) = cos x$
Power Rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ (used for the $x^{2}$ term)
Sum Rule: $\frac{d}{d x} (f (x) + g (x)) = f^{'} (x) + g^{'} (x)$

Summary of Steps

Identify components: Recognize $u = sin x$ and $v = x^{2} + sin x$ for the quotient rule
Differentiate separately: Find $u^{'} = cos x$ and $v^{'} = 2 x + cos x$
Apply quotient rule: Substitute into $\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$
Expand numerator: Distribute to get $x^{2} cos x + sin x cos x - 2 x sin x - sin x cos x$
Simplify: Cancel the $sin x cos x$ terms to obtain the final result $\frac{x ^{2} c o s x - 2 x s i n x}{( x ^{2} + s i n x ) ^{2}}$

2.6 Q-8

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.6 Q-8

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps