If `x=(t+1)/(t),y=(t-1)/(t)," then "(dy)/(dx)=`

Options

A. 1
B. 0
C. 2
D. `-1`

Answer

To find \(\frac{dy}{dx}\) given the equations \(x = \frac{t + 1}{t}\) and \(y = \frac{t - 1}{t}\), we will use the chain rule of differentiation. The steps are as follows:

### Step 1: Differentiate \(x\) with respect to \(t\)
Given:
\[ x = \frac{t + 1}{t} \]
We can rewrite \(x\) as:
\[ x = 1 + \frac{1}{t} \]
Now, we differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = 0 - \frac{1}{t^2} = -\frac{1}{t^2} \]

### Step 2: Differentiate \(y\) with respect to \(t\)
Given:
\[ y = \frac{t - 1}{t} \]
We can rewrite \(y\) as:
\[ y = 1 - \frac{1}{t} \]
Now, we differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = 0 + \frac{1}{t^2} = \frac{1}{t^2} \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the values we found:
\[ \frac{dy}{dx} = \frac{\frac{1}{t^2}}{-\frac{1}{t^2}} = -1 \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -1 \]

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