To find $\frac{dy}{dx}$ using implicit differentiation for the equation:

$5x^3 = -3xy + 2$

Step 1: Differentiate both sides with respect to $x$.

1. Differentiate the left side: $\frac{d}{dx}(5x^3) = 15x^2$

2. Differentiate the right side: Apply the product rule to the term $-3xy$ and differentiate $2$, which is constant:

   • The derivative of $-3xy$ using the product rule is: $\frac{d}{dx}(-3xy) = -3 \left( x \frac{d}{dx}(y) + y \frac{d}{dx}(x) \right) = -3(x \frac{dy}{dx} + y)$
   • The derivative of the constant $2$ is $0$.

So, the differentiated equation becomes: $15x^2 = -3(x \frac{dy}{dx} + y)$

Step 2: Solve for $\frac{dy}{dx}$.

Now, isolate $\frac{dy}{dx}$: $15x^2 = -3x \frac{dy}{dx} - 3y$ Add $3y$ to both sides: $15x^2 + 3y = -3x \frac{dy}{dx}$ Now, divide both sides by $-3x$: $\frac{dy}{dx} = \frac{-15x^2 - 3y}{-3x}$

Simplify: $\frac{dy}{dx} = \frac{15x^2 + 3y}{3x}$

Finally, factor out the common factor of 3: $\frac{dy}{dx} = \frac{5x^2 + y}{x}$

Final Answer:

$\frac{dy}{dx} = \frac{5x^2 + y}{x}$

Would you like more details on any part of this process? Here's a list of related questions to further explore:

1. What are the product and chain rules used for implicit differentiation?
2. How do you apply implicit differentiation to more complex equations?
3. What happens if the equation includes higher-order terms like $x^4$ or trigonometric functions?
4. How do you check if an implicit derivative is correct after differentiating?
5. What is the geometric interpretation of implicit differentiation in terms of slopes?

Tip: Remember that in implicit differentiation, when differentiating terms involving $y$, treat $y$ as a function of $x$, meaning you apply the chain rule ($\frac{d}{dx}y = \frac{dy}{dx}$).