https://file.aiquickdraw.com/mathbot/file/172785239050679gtbsj1.jpg

Let's go through the problems in the image step by step. Here’s a breakdown of each question:

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### 1. Determine the graph of the equation of the form:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]

This represents the general equation of a circle in standard form. We can rewrite it in a more recognizable form and identify the center and radius.

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### 2. Find \(\frac{dy}{dx}\) of \( y = \sqrt{x^2 - 4} \) using first principle formula.

This involves using the definition of the derivative from first principles:
\[
\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

We will apply this to \( y = \sqrt{x^2 - 4} \).

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### 3. Find the limit of the below functions:
- (i) \( \lim_{x \to 5} \frac{x^2 - 25}{x - 5} \)
- (ii) \( \lim_{x \to 0} \frac{\sqrt{x + 3} - \sqrt{3}}{x} \)
- (iii) \( \lim_{x \to 2} \left( \frac{1}{x-2} - \frac{4}{x^2 - 4} \right) \)

These are standard limit problems, potentially involving factorization, conjugates, or simplification techniques.

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### 4. Find the derivatives of:
- (i) \( D = \left( \frac{x+2}{x-3} \right)^3 \)
- (ii) \( J = \left( 4x^3 - 3x^2 \right)^2 (x+5)^3 \)

This involves applying the chain rule and product rule for derivatives.

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### 5. Given the function \( y = x^2 - 1 \) and \( x = \sqrt[3]{x^2 + 2} \), find \(\frac{dy}{dx}\).

This requires implicit differentiation since \( x \) is given in terms of itself.

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I will now begin solving these problems step by step. Would you like me to start with a specific problem, or should I go in order from 1 to 5? Let me know if you'd like any detailed explanation for each step!

1. Determine the graph of the equation of the form x^2 + y^2 + Dx + Ey + F = 0.
2. Find dy/dx of y = sqrt(x^2 - 4) using first principle formula.
3. Find the limit of the following functions:
   i) lim (x -> 5) ((x^2 - 25) / (x - 5))
   ii) lim (x -> 0) ((sqrt(x + 3) - sqrt(3)) / x)
   iii) lim (x -> 2) (1/(x-2) - 4/(x^2 - 4))
4. Find the derivatives of:
   i) D = ((x+2)/(x-3))^3
   ii) J = (4x^3 - 3x^2)^2 * (x+5)^3
   iii) Given y = x^2 - 1 and x = (x^2 + 2)^(1/3), find dy/dx.

This calculus assignment covers key topics such as graphing equations of circles, finding derivatives using first principles, solving limits of functions, and applying chain rule and product rule for differentiation. Suitable for students in higher-level calculus classes.

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