If `x=at,y=(a)/(t)," then "(dy)/(dx)=`

Options

A. `(1)/(t^(2))`
B. `t^(2)`
C. `-t^(2)`
D. `-(1)/(t^(2))`

Answer

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

### Step 1: Differentiate \(x\) with respect to \(t\)
Given:
\[ x = at \]
Differentiating both sides with respect to \(t\):
\[ \frac{dx}{dt} = a \]

### Step 2: Differentiate \(y\) with respect to \(t\)
Given:
\[ y = \frac{a}{t} = a t^{-1} \]
Differentiating both sides with respect to \(t\):
\[ \frac{dy}{dt} = -a t^{-2} = -\frac{a}{t^2} \]

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

### Step 4: Simplify the expression
The \(a\) in the numerator and denominator cancels out:
\[ \frac{dy}{dx} = -\frac{1}{t^2} \]

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

---