Question Statement

Find $\frac{d y}{d x}$ given that:

$y = \frac{5 - c o s x}{5 + s i n x}$

Background and Explanation

This problem requires differentiating a quotient of trigonometric functions using the quotient rule. You should be familiar with the basic derivatives $\frac{d}{d x} (sin x) = cos x$ and $\frac{d}{d x} (cos x) = - sin x$ , as well as the Pythagorean identity $sin^{2} x + cos^{2} x = 1$ .

Solution

We are given the function:

$y = \frac{5 - c o s x}{5 + s i n x}$

To find $\frac{d y}{d x}$ , we apply the quotient rule. For a function $y = \frac{u}{v}$ , the derivative is:

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

Let $u = 5 - cos x$ and $v = 5 + sin x$ .

First, compute the derivatives of the numerator and denominator:

$\frac{d}{d x} (5 - cos x) = 0 - (- sin x) = sin x$ (derivative of constant 5 is 0, derivative of $- cos x$ is $sin x$ )
$\frac{d}{d x} (5 + sin x) = 0 + cos x = cos x$ (derivative of constant 5 is 0, derivative of $sin x$ is $cos x$ )

Now substitute into the quotient rule formula:

$\frac{d y}{d x} = \frac{( 5 + sin x ) ( 0 + sin x ) - ( 5 - cos x ) ( 0 + cos x )}{( 5 + sin x ) ^{2}} = \frac{( 5 + sin x ) ( sin x ) - ( 5 - cos x ) ( cos x )}{( 5 + sin x ) ^{2}} = \frac{5 sin x + sin ^{2} x - 5 cos x + cos ^{2} x}{( 5 + sin x ) ^{2}} = \frac{5 sin x - 5 cos x + sin ^{2} x + cos ^{2} x}{( 5 + sin x ) ^{2}} = \frac{5 sin x - 5 cos x + 1}{( 5 + sin x ) ^{2}}$

Step-by-step reasoning:

In the first line, we substitute the derivatives into the quotient rule, explicitly showing the derivative of constants as $0$ .
In the second line, we simplify the expressions $(0 + sin x)$ to $sin x$ and $(0 + cos x)$ to $cos x$ .
In the third line, we expand the numerator: $(5 + sin x) sin x = 5 sin x + sin^{2} x$ and $- (5 - cos x) cos x = - 5 cos x + cos^{2} x$ (note the sign change: minus times minus gives plus for the $cos^{2} x$ term).
In the fourth line, we rearrange terms to group $sin^{2} x$ and $cos^{2} x$ together.
In the final line, we apply the Pythagorean identity $sin^{2} x + cos^{2} x = 1$ .

Thus, the derivative is:

$\frac{d y}{d x} = \frac{5 s i n x - 5 c o s x + 1}{( 5 + s i n x ) ^{2}}$

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: $\frac{d}{d x} (sin x) = cos x$
Derivative of cosine: $\frac{d}{d x} (cos x) = - sin x$
Constant rule: $\frac{d}{d x} (c) = 0$
Pythagorean Identity: $sin^{2} x + cos^{2} x = 1$

Summary of Steps

Identify $u = 5 - cos x$ and $v = 5 + sin x$ from the given rational function
Calculate derivatives: $u^{'} = sin x$ and $v^{'} = cos x$ (remembering that $\frac{d}{d x} (cos x) = - sin x$ )
Apply the quotient rule: $\frac{v \cdot u ^{'} - u \cdot v ^{'}}{v ^{2}}$
Expand the numerator carefully, watching signs: $5 sin x + sin^{2} x - 5 cos x + cos^{2} x$
Rearrange terms to group $sin^{2} x + cos^{2} x$ and apply the identity to replace with $1$
Write final answer: $\frac{5 s i n x - 5 c o s x + 1}{( 5 + s i n x ) ^{2}}$

Solution

We are given the function:

y = \frac{5 - c o s x}{5 + s i n x}

To find

\frac{d y}{d x}

, we apply the quotient rule. For a function

y = \frac{u}{v}

, the derivative is:

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

Let

u = 5 - cos x

and

v = 5 + sin x

First, compute the derivatives of the numerator and denominator:

\frac{d}{d x} (5 - cos x) = 0 - (- sin x) = sin x

(derivative of constant 5 is 0, derivative of

- cos x

sin x

)

\frac{d}{d x} (5 + sin x) = 0 + cos x = cos x

(derivative of constant 5 is 0, derivative of

sin x

cos x

)

Now substitute into the quotient rule formula:

\frac{d y}{d x} = \frac{( 5 + sin x ) ( 0 + sin x ) - ( 5 - cos x ) ( 0 + cos x )}{( 5 + sin x ) ^{2}} = \frac{( 5 + sin x ) ( sin x ) - ( 5 - cos x ) ( cos x )}{( 5 + sin x ) ^{2}} = \frac{5 sin x + sin ^{2} x - 5 cos x + cos ^{2} x}{( 5 + sin x ) ^{2}} = \frac{5 sin x - 5 cos x + sin ^{2} x + cos ^{2} x}{( 5 + sin x ) ^{2}} = \frac{5 sin x - 5 cos x + 1}{( 5 + sin x ) ^{2}}

Step-by-step reasoning:

In the first line, we substitute the derivatives into the quotient rule, explicitly showing the derivative of constants as

0

In the second line, we simplify the expressions

(0 + sin x)

sin x

and

(0 + cos x)

cos x

In the third line, we expand the numerator:

(5 + sin x) sin x = 5 sin x + sin^{2} x

and

- (5 - cos x) cos x = - 5 cos x + cos^{2} x

(note the sign change: minus times minus gives plus for the

cos^{2} x

term).

In the fourth line, we rearrange terms to group

sin^{2} x

and

cos^{2} x

together.

In the final line, we apply the Pythagorean identity

sin^{2} x + cos^{2} x = 1

Thus, the derivative is:

\frac{d y}{d x} = \frac{5 s i n x - 5 c o s x + 1}{( 5 + s i n x ) ^{2}}

Summary of Steps

Identify

u = 5 - cos x

and

v = 5 + sin x

from the given rational function

Calculate derivatives:

u^{'} = sin x

and

v^{'} = cos x

(remembering that

\frac{d}{d x} (cos x) = - sin x

)

Apply the quotient rule:

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

Expand the numerator carefully, watching signs:

5 sin x + sin^{2} x - 5 cos x + cos^{2} x

Rearrange terms to group

sin^{2} x + cos^{2} x

and apply the identity to replace with

1

Write final answer:

\frac{5 s i n x - 5 c o s x + 1}{( 5 + s i n x ) ^{2}}

2.6 Q-6

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.6 Q-6

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps