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

Answer

To find \(\frac{dy}{dx}\) when \(x = 4t\) and \(y = \frac{4}{t}\), we can use the chain rule of differentiation. Here are the steps to solve the problem:

### Step 1: Differentiate \(x\) with respect to \(t\)
Given \(x = 4t\), we differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = 4 \]

**Hint:** Remember that the derivative of a constant multiplied by a variable is just the constant.

### Step 2: Differentiate \(y\) with respect to \(t\)
Given \(y = \frac{4}{t}\), we differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(4t^{-1}) = -4t^{-2} = -\frac{4}{t^2} \]

**Hint:** Use the power rule for differentiation, where \(\frac{d}{dt}(t^n) = nt^{n-1}\).

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)
Now we can find \(\frac{dy}{dx}\) using the relationship:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{-\frac{4}{t^2}}{4} \]

### Step 4: Simplify the expression
Now simplify the expression:
\[ \frac{dy}{dx} = -\frac{4}{t^2} \cdot \frac{1}{4} = -\frac{1}{t^2} \]

### Final Result
Thus, the final result is:
\[ \frac{dy}{dx} = -\frac{1}{t^2} \]

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