Find `(dy)/(dx)` when x=4t , y=`4/t`

Answer

To find \(\frac{dy}{dx}\) when \(x = 4t\) and \(y = \frac{4}{t}\), we will use the chain rule. We can express \(\frac{dy}{dx}\) in terms of \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) as follows:

\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]

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

Given \(y = \frac{4}{t}\), we can rewrite this as:

\[ y = 4t^{-1} \]

Now, we differentiate \(y\) with respect to \(t\):

\[ \frac{dy}{dt} = 4 \cdot (-1) \cdot t^{-2} = -\frac{4}{t^2} \]

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

Given \(x = 4t\), we differentiate \(x\) with respect to \(t\):

\[ \frac{dx}{dt} = 4 \]

### Step 3: Substitute \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) into the chain rule formula

Now we can substitute the values we found into the formula for \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-\frac{4}{t^2}}{4} \]

### Step 4: Simplify the expression

Now we simplify the expression:

\[ \frac{dy}{dx} = -\frac{4}{t^2} \cdot \frac{1}{4} = -\frac{1}{t^2} \]

### Final Result

Thus, the derivative \(\frac{dy}{dx}\) when \(x = 4t\) and \(y = \frac{4}{t}\) is:

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