Implicit Differentiation Exam Question

Consider the equation:

1. Use implicit differentiation to find $\frac{{dy}}{{dx}}$ in terms of $x$ and $y$.

Solution:

To find $\frac{{dy}}{{dx}}$ using implicit differentiation, we differentiate both sides of the given equation with respect to $x$.

Differentiating each term separately, we get:

Using the power rule, chain rule, and product rule, the derivatives are:

Now, we isolate $\frac{{dy}}{{dx}}$ term:

Factor out $\frac{{dy}}{{dx}}$:

Finally, divide both sides by $3y^2 - 3x$ to solve for $\frac{{dy}}{{dx}}$:

Hence, $\frac{{dy}}{{dx}}$ in terms of $x$ and $y$ is:

Therefore, the answer is: [ \frac{{dy}}{{dx}} = \frac{{-3x^2 + 3y}}{{3y^2 - 3x}} ]