Find `dy/dx`, if x and y are connected parametrically by the equations, given below without eliminating the parameter:
`x=logt,y=sint`

Answer

To find \(\frac{dy}{dx}\) given the parametric equations \(x = \log t\) and \(y = \sin t\), we will follow these steps:

### Step 1: Differentiate \(x\) with respect to \(t\)
We start by differentiating \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(\log t) = \frac{1}{t} \]

### Step 2: Differentiate \(y\) with respect to \(t\)
Next, we differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(\sin t) = \cos t \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)
Now, we can find \(\frac{dy}{dx}\) using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\cos t}{\frac{1}{t}} = t \cdot \cos t \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = t \cos t \]

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