Differentiate each of the following with respect to x in following question: `(x^(5)-cosx)/(sinx)`

Answer

To differentiate the function \( y = \frac{x^5 - \cos x}{\sin x} \) with respect to \( x \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = x^5 - \cos x \) and \( v = \sin x \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x^5 - \cos x \)
- \( v = \sin x \)

### Step 2: Differentiate \( u \) and \( v \)
Now we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

1. Differentiate \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^5) - \frac{d}{dx}(\cos x) = 5x^4 + \sin x \]

2. Differentiate \( v \):
\[ \frac{dv}{dx} = \frac{d}{dx}(\sin x) = \cos x \]

### Step 3: Apply the Quotient Rule
Now we apply the quotient rule:

\[ \frac{dy}{dx} = \frac{\sin x (5x^4 + \sin x) - (x^5 - \cos x)(\cos x)}{\sin^2 x} \]

### Step 4: Simplify the Expression
Now we simplify the numerator:

1. Expand the numerator:
\[ \sin x (5x^4 + \sin x) = 5x^4 \sin x + \sin^2 x \]
\[ (x^5 - \cos x)(\cos x) = x^5 \cos x - \cos^2 x \]

2. Combine the terms:
\[ \text{Numerator} = 5x^4 \sin x + \sin^2 x - (x^5 \cos x - \cos^2 x) \]
\[ = 5x^4 \sin x + \sin^2 x - x^5 \cos x + \cos^2 x \]

Thus, the derivative can be expressed as:

\[ \frac{dy}{dx} = \frac{5x^4 \sin x + \sin^2 x - x^5 \cos x + \cos^2 x}{\sin^2 x} \]

### Final Result
The final result for the derivative of the function \( y = \frac{x^5 - \cos x}{\sin x} \) is:

\[ \frac{dy}{dx} = \frac{5x^4 \sin x - x^5 \cos x + \sin^2 x + \cos^2 x}{\sin^2 x} \]