can i please get the derivative for this function? y=0.029(x-670)^2+94

To find the derivative of the function \( y = 0.029(x - 670)^2 + 94 \) with respect to \( x \), follow these steps:

Given:
\[
y = 0.029(x - 670)^2 + 94
\]

We can differentiate each part of the function with respect to \( x \):

1. The derivative of the term \( 0.029(x - 670)^2 \):
   - The outer function is \( u^2 \), where \( u = x - 670 \).
   - The derivative of \( u^2 \) with respect to \( u \) is \( 2u \).
   - Multiply by the derivative of \( u = x - 670 \) with respect to \( x \), which is \( 1 \).

So, the derivative of \( 0.029(x - 670)^2 \) is:
\[
\frac{d}{dx}\left[0.029(x - 670)^2\right] = 0.029 \times 2(x - 670) = 0.058(x - 670)
\]

2. The derivative of the constant term \( 94 \) is \( 0 \).

Thus, the derivative of the function is:
\[
\frac{dy}{dx} = 0.058(x - 670)
\]

Would you like any further details or explanations?

Here are five related questions for further exploration:

1. How does the derivative change if the coefficient 0.029 is modified?
2. What is the significance of the point \( x = 670 \) in this function?
3. How would you find the second derivative of the function \( y = 0.029(x - 670)^2 + 94 \)?
4. What is the slope of the tangent line to the curve at \( x = 670 \)?
5. How does the graph of this quadratic function relate to its derivative?

**Tip:** When dealing with quadratic functions, the derivative is a linear function, which represents the slope of the tangent line at any point on the curve.

Learn how to find the derivative of the function y = 0.029(x - 670)^2 + 94 with a step-by-step solution. This problem involves differentiation of quadratic functions and application of the power rule of differentiation. Suitable for Grades 11-12 students and anyone interested in calculus.

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