derive this function y=0.000189554273x^{3}-0.138976752x^{2}+32.8379259x-2287.44929

To derive the given function with respect to \( x \), we will apply the power rule to each term of the function. The function is:

\[
y = 0.000189554273x^{3} - 0.138976752x^{2} + 32.8379259x - 2287.44929
\]

### Step-by-Step Derivation:

1. **Differentiate the first term:**
   \[
   \frac{d}{dx}\left(0.000189554273x^{3}\right) = 0.000189554273 \times 3x^{2} = 0.000568662819x^{2}
   \]

2. **Differentiate the second term:**
   \[
   \frac{d}{dx}\left(-0.138976752x^{2}\right) = -0.138976752 \times 2x = -0.277953504x
   \]

3. **Differentiate the third term:**
   \[
   \frac{d}{dx}\left(32.8379259x\right) = 32.8379259
   \]

4. **Differentiate the constant term:**
   \[
   \frac{d}{dx}\left(-2287.44929\right) = 0
   \]

### Derivative of the Function:
Combining these results, the derivative of the function is:

\[
\frac{dy}{dx} = 0.000568662819x^{2} - 0.277953504x + 32.8379259
\]

Would you like more details on any specific part of this derivation, or do you have any other questions?

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Here are 5 related questions for further exploration:

1. How do you find the second derivative of the given function?
2. What does the derivative represent in the context of this function?
3. How would the derivative change if the function had a higher degree, like \(x^4\)?
4. What are the critical points of this function?
5. How can you determine if the function has any inflection points?

**Tip:** Remember that the derivative of a polynomial function is always a polynomial with one degree lower than the original function.

Learn how to derive the function y = 0.000189554273x^3 - 0.138976752x^2 + 32.8379259x - 2287.44929 with step-by-step differentiation. This problem covers differentiation of polynomial functions using the power rule. Suitable for students in Grades 11-12.

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