Let's break down the three problems in the image.

Problem 1:

Given the equation $4x^9 + y^7 = 3x$:

(a) Find $y'$ by implicit differentiation.

To differentiate implicitly:

• Derivative of $4x^9$ is $36x^8$,
• Derivative of $y^7$ with respect to $x$ is $7y^6 \cdot \frac{dy}{dx}$,
• Derivative of $3x$ is 3.

So: $36x^8 + 7y^6 \cdot \frac{dy}{dx} = 3$ Solve for $\frac{dy}{dx}$: $7y^6 \cdot \frac{dy}{dx} = 3 - 36x^8$ $\frac{dy}{dx} = \frac{3 - 36x^8}{7y^6}$

(b) Solve the equation explicitly for $y$ and differentiate to get $y'$ in terms of $x$.

Solving for $y$ in terms of $x$ is not straightforward, but differentiating implicitly as in part (a) is the correct approach for obtaining $y'$.

(c) Check consistency by substituting the explicit expression into the implicit derivative.

You would substitute the value of $y$ (from solving the equation in part (b)) back into the implicit derivative.

Problem 2:

Given the equation $x^2 - 10xy + y^2 = 10$:

We apply implicit differentiation:

• Derivative of $x^2$ is $2x$,
• Derivative of $-10xy$ (using the product rule) is $-10(y + x\frac{dy}{dx})$,
• Derivative of $y^2$ is $2y\frac{dy}{dx}$.

So the equation becomes: $2x - 10(y + x \cdot \frac{dy}{dx}) + 2y \cdot \frac{dy}{dx} = 0$ Simplify and solve for $\frac{dy}{dx}$.

Problem 3:

Given $x^5 - xy^4 + y^5 = 1$:

Implicit differentiation yields:

• Derivative of $x^5$ is $5x^4$,
• Derivative of $-xy^4$ using the product rule is $-(y^4 + 4xy^3\frac{dy}{dx})$,
• Derivative of $y^5$ is $5y^4\frac{dy}{dx}$.

So: $5x^4 - (y^4 + 4xy^3 \cdot \frac{dy}{dx}) + 5y^4 \cdot \frac{dy}{dx} = 0$ Solve for $\frac{dy}{dx}$.

Would you like detailed steps for each solution? Here are five follow-up questions:

1. How does implicit differentiation work when both $x$ and $y$ are involved in a product?
2. What is the product rule, and how is it applied in implicit differentiation?
3. Can you always solve for $y$ explicitly before differentiating?
4. What conditions make solving for $y$ explicitly impossible or impractical?
5. How does substituting solutions verify the consistency of implicit differentiation?

Tip: When using implicit differentiation, remember to apply the chain rule whenever differentiating a function of $y$ with respect to $x$.