$$ y=x^2+5x+3,\frac{\differentialD y}{d\left(x^2\right)}= $$

To find $ \frac{d y}{d(x^2)} $, we need to first differentiate the given function $ y = x^2 + 5x + 3 $ with respect to $ x^2 $. This involves applying the chain rule.

### Step-by-Step Process:

1. The chain rule states that:
   \[
   \frac{d y}{d(x^2)} = \frac{d y}{dx} \cdot \frac{dx}{d(x^2)}
   \]
2. First, differentiate $ y = x^2 + 5x + 3 $ with respect to $ x $:
   \[
   \frac{d y}{dx} = 2x + 5
   \]
3. Next, differentiate $ x $ with respect to $ x^2 $:
   \[
   \frac{dx}{d(x^2)} = \frac{1}{2x}
   \]
4. Now, apply the chain rule:
   \[
   \frac{d y}{d(x^2)} = (2x + 5) \cdot \frac{1}{2x}
   \]
5. Simplify the expression:
   \[
   \frac{d y}{d(x^2)} = \frac{2x + 5}{2x}
   \]

Thus, the derivative is:
\[
\frac{d y}{d(x^2)} = 1 + \frac{5}{2x}
\]

Would you like further details or have any questions about the steps?

Here are 5 related questions for practice:
1. How would you differentiate $ y = 3x^2 + 4x + 1 $ with respect to $ x^2 $?
2. What is the derivative of $ y = x^3 + 2x^2 + x $ with respect to $ x^2 $?
3. How would you apply the chain rule if $ y = e^{x^2} $?
4. What is the derivative of $ y = \sin(x^2) $ with respect to $ x^2 $?
5. How would you simplify $ \frac{(3x + 7)}{2x} $?

**Tip**: When using the chain rule, always make sure to identify both the inner and outer functions properly to avoid mistakes.

Learn how to apply the chain rule to differentiate y = x^2 + 5x + 3 with respect to x^2. This problem uses the chain rule and basic differentiation techniques. Suitable for students in Grades 10-12 studying algebra and calculus.

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