Find $\frac{d y}{d x}$ by implicit differentiation for the following equation. \[ e^{x^{2} y}=6 x+4 y+7 \] \[ \frac{d y}{d x}= \]

Step 1 ：First, we need to apply the chain rule to differentiate the left side of the equation. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is \(e^u\) and the inner function is \(x^2y\). The derivative of \(e^u\) is \(e^u\) and the derivative of \(x^2y\) is \(2xy + x^2 \frac{dy}{dx}\).

Step 2 ：So, the derivative of the left side of the equation is \((2xy + x^2 \frac{dy}{dx})e^{x^2y}\).

Step 3 ：Next, we differentiate the right side of the equation. The derivative of \(6x\) is \(6\), the derivative of \(4y\) is \(4 \frac{dy}{dx}\), and the derivative of \(7\) is \(0\).

Step 4 ：So, the derivative of the right side of the equation is \(6 + 4 \frac{dy}{dx}\).

Step 5 ：Setting these two derivatives equal to each other, we get the equation \((2xy + x^2 \frac{dy}{dx})e^{x^2y} = 6 + 4 \frac{dy}{dx}\).

Step 6 ：Next, we isolate \(\frac{dy}{dx}\) on one side of the equation. To do this, we first subtract \(4 \frac{dy}{dx}\) from both sides to get \((2xy + x^2 \frac{dy}{dx})e^{x^2y} - 4 \frac{dy}{dx} = 6\).

Step 7 ：Then, we factor out \(\frac{dy}{dx}\) to get \(\frac{dy}{dx}(x^2e^{x^2y} - 4) = 6 - 2xye^{x^2y}\).

Step 8 ：Finally, we divide both sides by \(x^2e^{x^2y} - 4\) to solve for \(\frac{dy}{dx}\). This gives us \(\frac{dy}{dx} = \frac{6 - 2xye^{x^2y}}{x^2e^{x^2y} - 4}\).

Step 9 ：Therefore, the derivative of \(y\) with respect to \(x\) is \(\boxed{\frac{6 - 2xye^{x^2y}}{x^2e^{x^2y} - 4}}\).