https://file.aiquickdraw.com/mathbot/file/1730227930217r7v268ig.jpg

Let's solve each part of the problem based on the given equation:

\[
5x^2 - y^2 = 6
\]

### Part (a): Find \( y' \) by implicit differentiation.

1. Differentiate both sides of the equation with respect to \( x \):
   \[
   \frac{d}{dx}(5x^2 - y^2) = \frac{d}{dx}(6)
   \]

2. Apply the derivative term-by-term:
   \[
   10x - 2y \cdot y' = 0
   \]

3. Solve for \( y' \):
   \[
   y' = \frac{10x}{2y} = \frac{5x}{y}
   \]

So, for part (a):
\[
y' = \frac{5x}{y}
\]

### Part (b): Solve the equation explicitly for \( y \) and differentiate to get \( y' \) in terms of \( x \).

1. Solve the original equation for \( y \):
   \[
   y^2 = 5x^2 - 6
   \]
   \[
   y = \pm \sqrt{5x^2 - 6}
   \]

2. Differentiate \( y = \sqrt{5x^2 - 6} \) with respect to \( x \) (taking only the positive root for simplicity):
   \[
   y' = \frac{d}{dx}\left(\sqrt{5x^2 - 6}\right)
   \]

3. Using the chain rule:
   \[
   y' = \frac{1}{2\sqrt{5x^2 - 6}} \cdot (10x) = \frac{5x}{\sqrt{5x^2 - 6}}
   \]

Thus, for part (b):
\[
y' = \pm \frac{5x}{\sqrt{5x^2 - 6}}
\]

---

Would you like further details on any of the steps? Here are some related questions that might help deepen your understanding:

1. Why do we use implicit differentiation in part (a) instead of solving for \( y \) directly?
2. What are the differences between implicit and explicit differentiation?
3. How do we apply the chain rule in implicit differentiation?
4. What does the \( \pm \) sign represent in the explicit solution for \( y \)?
5. How would the process change if we had an equation like \( 5x^2 - y^2 = c \) for a general constant \( c \)?

**Tip:** When differentiating implicitly, always remember to apply the chain rule carefully to terms involving \( y \), as \( y \) is treated as a function of \( x \).

Consider the following equation: 5x^2 - y^2 = 6.
(a) Find y' by implicit differentiation.
(b) Solve the equation explicitly for y and differentiate to get y' in terms of x.

Solve for y' using implicit differentiation and find y' explicitly for the equation 5x^2 - y^2 = 6. This problem demonstrates the use of the chain rule and both implicit and explicit differentiation methods, ideal for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation