The image contains an assignment on implicit differentiation from Calculus I. Let's solve these problems one by one.

Problem 1:

$5 = x^2 + y^2$ To find $\frac{dy}{dx}$, differentiate both sides implicitly with respect to $x$:

$0 = 2x + 2y \frac{dy}{dx}$ Solving for $\frac{dy}{dx}$: $2y \frac{dy}{dx} = -2x$ $\frac{dy}{dx} = \frac{-x}{y}$

Problem 2:

$x^3 + y^3 - 3axy = 0$ Differentiating implicitly with respect to $x$:

$3x^2 + 3y^2 \frac{dy}{dx} - 3a\left( y + x \frac{dy}{dx} \right) = 0$ Expanding and solving for $\frac{dy}{dx}$: $3x^2 + 3y^2 \frac{dy}{dx} - 3ay - 3ax \frac{dy}{dx} = 0$ Grouping terms with $\frac{dy}{dx}$: $3y^2 \frac{dy}{dx} - 3ax \frac{dy}{dx} = -3x^2 + 3ay$ Factoring: $\frac{dy}{dx}(3y^2 - 3ax) = 3ay - 3x^2$ $\frac{dy}{dx} = \frac{3ay - 3x^2}{3y^2 - 3ax}$

Problem 3:

$y^2(x + 2y) = c^3$ Differentiate implicitly with respect to $x$: $2y \frac{dy}{dx}(x + 2y) + y^2 \left(1 + 2\frac{dy}{dx}\right) = 0$ Expanding: $2y(x + 2y)\frac{dy}{dx} + y^2 + 2y^2 \frac{dy}{dx} = 0$ Solving for $\frac{dy}{dx}$: $\frac{dy}{dx} \left( 2y(x + 2y) + 2y^2 \right) = -y^2$ $\frac{dy}{dx} = \frac{-y^2}{2y(x + 2y) + 2y^2}$

Problem 4:

$x^2 + 4y^2 = 4xy$ Differentiate both sides: $2x + 8y \frac{dy}{dx} = 4\left( y + x \frac{dy}{dx} \right)$ Expanding: $2x + 8y \frac{dy}{dx} = 4y + 4x \frac{dy}{dx}$ Rearranging: $8y \frac{dy}{dx} - 4x \frac{dy}{dx} = 4y - 2x$ Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(8y - 4x) = 4y - 2x$ $\frac{dy}{dx} = \frac{4y - 2x}{8y - 4x}$

Problem 5:

$(x + y)^2 = 2ay$ Differentiate implicitly: $2(x + y)(1 + \frac{dy}{dx}) = 2a \frac{dy}{dx}$ Simplify: $(x + y)(1 + \frac{dy}{dx}) = a \frac{dy}{dx}$ Expanding: $x + y + (x + y)\frac{dy}{dx} = a \frac{dy}{dx}$ Rearranging: $(x + y)\frac{dy}{dx} - a \frac{dy}{dx} = -x - y$ Factor out $\frac{dy}{dx}$: $\frac{dy}{dx} \left( (x + y) - a \right) = -x - y$ $\frac{dy}{dx} = \frac{-x - y}{x + y - a}$

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Here are 5 related questions:

1. How is implicit differentiation different from explicit differentiation?
2. Can you apply implicit differentiation to any type of function, or are there specific cases?
3. What is the significance of the chain rule in implicit differentiation?
4. How would you verify if an implicit solution is correct for a given equation?
5. How does the concept of partial derivatives relate to implicit differentiation?

Tip:

Always remember to apply the product and chain rules carefully when differentiating terms with multiple variables in implicit differentiation.