Find `dy/dx if y=e^x/sinx`

Answer

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

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

where \(u = e^x\) and \(v = \sin x\).

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

### Step 2: Differentiate \(u\) and \(v\)
Now we need to find the derivatives of \(u\) and \(v\):
- \(\frac{du}{dx} = e^x\) (the derivative of \(e^x\) is \(e^x\))
- \(\frac{dv}{dx} = \cos x\) (the derivative of \(\sin x\) is \(\cos x\))

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

\[ \frac{dy}{dx} = \frac{\sin x \cdot e^x - e^x \cdot \cos x}{(\sin x)^2} \]

### Step 4: Simplify the Expression
We can factor out \(e^x\) from the numerator:

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

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:

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

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