Find ` (dy)/(dx) , if x = c t, y= ( c )/( t) `

Answer

To find \(\frac{dy}{dx}\) given \(x = ct\) and \(y = \frac{c}{t}\), we can use the chain rule. Here’s a step-by-step solution:

### Step 1: Differentiate \(y\) with respect to \(t\)

Given:
\[ y = \frac{c}{t} \]

We differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}\left(\frac{c}{t}\right) = -\frac{c}{t^2} \]

### Step 2: Differentiate \(x\) with respect to \(t\)

Given:
\[ x = ct \]

We differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = c \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)

Using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} \]

We already have \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\). To find \(\frac{dt}{dx}\), we take the reciprocal of \(\frac{dx}{dt}\):
\[ \frac{dt}{dx} = \frac{1}{\frac{dx}{dt}} = \frac{1}{c} \]

Now substituting the values:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} = \left(-\frac{c}{t^2}\right) \cdot \left(\frac{1}{c}\right) \]

### Step 4: Simplify the expression

\[ \frac{dy}{dx} = -\frac{c}{t^2} \cdot \frac{1}{c} = -\frac{1}{t^2} \]

### Final Result

Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{1}{t^2} \]

---