The image contains mathematical derivative formulas and examples for calculating derivatives of functions. Let’s solve the three examples step by step using the given rules:

Example 1: $Y = 5(x^2 + 3)^3$

Using the chain rule:

1. Let $u = x^2 + 3$, so $Y = 5u^3$.
2. Derivative of $5u^3$: $Y' = 5 \cdot 3u^2 \cdot u'$.
3. $u' = 2x$ (derivative of $x^2 + 3$).
4. Substituting $u = x^2 + 3$: $Y' = 5 \cdot 3(x^2 + 3)^2 \cdot 2x = 30x(x^2 + 3)^2.$

Example 2: $Y = (x^3 + 2)^2(x^2 + 5)^5$

Using the product rule:

1. Let $u = (x^3 + 2)^2$ and $v = (x^2 + 5)^5$.
2. Derivative of $u$: $u' = 2(x^3 + 2) \cdot 3x^2 = 6x^2(x^3 + 2)$.
3. Derivative of $v$: $v' = 5(x^2 + 5)^4 \cdot 2x = 10x(x^2 + 5)^4$.
4. Using $Y' = u'v + uv'$: [ Y' = [6x^2(x^3 + 2) \cdot (x^2 + 5)^5] + [(x^3 + 2)^2 \cdot 10x(x^2 + 5)^4]. ]

Example 3: $Y = \frac{x^2 + 2x}{x^3 + 2}$

Using the quotient rule:

1. Let $u = x^2 + 2x$ and $v = x^3 + 2$.
2. Derivative of $u$: $u' = 2x + 2$.
3. Derivative of $v$: $v' = 3x^2$.
4. Using $Y' = \frac{u'v - uv'}{v^2}$: $Y' = \frac{(2x + 2)(x^3 + 2) - (x^2 + 2x)(3x^2)}{(x^3 + 2)^2}.$ Simplify numerator: $(2x + 2)(x^3 + 2) = 2x^4 + 4x + 2x^3 + 4,$ $(x^2 + 2x)(3x^2) = 3x^4 + 6x^3.$ Subtract: $Y' = \frac{(2x^4 + 2x^3 + 4x + 4) - (3x^4 + 6x^3)}{(x^3 + 2)^2},$ $Y' = \frac{-x^4 - 4x^3 + 4x + 4}{(x^3 + 2)^2}.$

Would you like me to simplify further, or explain any specific steps?

Related Questions:

1. How can the chain rule be applied in more complex cases?
2. What are the applications of product rule derivatives in real-world problems?
3. How does the quotient rule help in simplifying complex fraction derivatives?
4. What happens when functions involve logarithms or trigonometric terms?
5. Can you show the graphical interpretation of these derivative results?

Tip:

Always simplify your final derivative expressions to make them easier