Given the parametic equations ` x= f (t) , y= g(t) ` then ` (d^2 y)/(dx^2)` equals

Options

A. `((d^2 y)/(dt^2) . (dx)/(dt) - (dy)/(dt)- (dy)/(dt) (d^2 x)/(dt^2))/((dx // dt)^2)`
B. `((dx)/(dt)(d^2 y)/(dt^2) - (d^2 x )/(dt^2 ) (dy)/(dt))/((dx //dt)^3)`
C. `(d^2y )/(dt^2) // (d^2 x)/(dx^2)`
D. None

Answer

To find the second derivative \(\frac{d^2y}{dx^2}\) given the parametric equations \(x = f(t)\) and \(y = g(t)\), we can follow these steps:

### Step 1: Find the first derivative \(\frac{dy}{dx}\)

Using the chain rule for parametric equations, we can express \(\frac{dy}{dx}\) as:

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

### Step 2: Differentiate \(\frac{dy}{dx}\) with respect to \(x\)

To find \(\frac{d^2y}{dx^2}\), we need to differentiate \(\frac{dy}{dx}\) with respect to \(x\). This is done using the chain rule again:

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) \]

Using the chain rule, we can express this as:

\[ \frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} \]

### Step 3: Substitute \(\frac{dy}{dx}\) into the differentiation

Substituting \(\frac{dy}{dx}\) from Step 1, we have:

\[ \frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\right) \cdot \frac{dt}{dx} \]

### Step 4: Apply the quotient rule

Now we apply the quotient rule to differentiate \(\frac{dy}{dx}\):

\[ \frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{\frac{d^2y}{dt^2} \cdot \frac{dx}{dt} - \frac{dy}{dt} \cdot \frac{d^2x}{dt^2}}{\left(\frac{dx}{dt}\right)^2} \]

### Step 5: Substitute back into the expression for \(\frac{d^2y}{dx^2}\)

Now substituting this back into our expression for \(\frac{d^2y}{dx^2}\):

\[ \frac{d^2y}{dx^2} = \left(\frac{\frac{d^2y}{dt^2} \cdot \frac{dx}{dt} - \frac{dy}{dt} \cdot \frac{d^2x}{dt^2}}{\left(\frac{dx}{dt}\right)^2}\right) \cdot \frac{1}{\frac{dx}{dt}} \]

### Step 6: Simplify the expression

This simplifies to:

\[ \frac{d^2y}{dx^2} = \frac{d^2y/dt^2 \cdot dx/dt - dy/dt \cdot d^2x/dt^2}{\left(\frac{dx}{dt}\right)^3} \]

### Final Result

Thus, the final expression for the second derivative \(\frac{d^2y}{dx^2}\) is:

\[ \frac{d^2y}{dx^2} = \frac{\frac{d^2y}{dt^2} \cdot \frac{dx}{dt} - \frac{dy}{dt} \cdot \frac{d^2x}{dt^2}}{\left(\frac{dx}{dt}\right)^3} \]

---