Find `(dy)/(dx)" when "y=(x^(5)-cosx)/(sinx)`.

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \frac{x^5 - \cos x}{\sin x}\), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form of \(\frac{a}{b}\), then the derivative is given by:

\[ \frac{dy}{dx} = \frac{b \frac{da}{dx} - a \frac{db}{dx}}{b^2} \]

where \(a = x^5 - \cos x\) and \(b = \sin x\).

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

### Step 2: Differentiate \(a\) and \(b\)
Now, we need to find \(\frac{da}{dx}\) and \(\frac{db}{dx}\).

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

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

### Step 3: Apply the Quotient Rule
Now, we can apply the quotient rule:
\[ \frac{dy}{dx} = \frac{\sin x \cdot (5x^4 + \sin x) - (x^5 - \cos x) \cdot \cos x}{(\sin x)^2} \]

### Step 4: Simplify the Expression
Now, we will simplify the numerator:
1. Expand the terms:
\[ \sin x \cdot (5x^4 + \sin x) = 5x^4 \sin x + \sin^2 x \]
\[ (x^5 - \cos x) \cdot \cos x = x^5 \cos x - \cos^2 x \]

2. Substitute back into the derivative:
\[ \frac{dy}{dx} = \frac{5x^4 \sin x + \sin^2 x - (x^5 \cos x - \cos^2 x)}{(\sin x)^2} \]

3. Combine like terms:
\[ = \frac{5x^4 \sin x + \sin^2 x - x^5 \cos x + \cos^2 x}{(\sin x)^2} \]

### Step 5: Final Simplification
Notice that \(\sin^2 x + \cos^2 x = 1\):
\[ \frac{dy}{dx} = \frac{5x^4 \sin x - x^5 \cos x + 1}{(\sin x)^2} \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{5x^4 \sin x - x^5 \cos x + 1}{\sin^2 x} \]