If `x = 2t^3` and `y = 3t^2`, then value of `(dy)/(dx)` is

Options

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

Answer

To find the value of \(\frac{dy}{dx}\) given the equations \(x = 2t^3\) and \(y = 3t^2\), we will follow these steps:

### Step 1: Differentiate \(y\) with respect to \(t\)
We start with the equation for \(y\):
\[ y = 3t^2 \]
To find \(\frac{dy}{dt}\), we differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(3t^2) = 3 \cdot 2t^{2-1} = 6t \]

### Step 2: Differentiate \(x\) with respect to \(t\)
Next, we differentiate \(x\) with respect to \(t\):
\[ x = 2t^3 \]
To find \(\frac{dx}{dt}\), we differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(2t^3) = 2 \cdot 3t^{3-1} = 6t^2 \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)
Now we can use the chain rule to find \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{6t}{6t^2} \]

### Step 4: Simplify the expression
We simplify the expression:
\[ \frac{dy}{dx} = \frac{6t}{6t^2} = \frac{1}{t} \]

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

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