Questions: For the function y=f(x)=-2 x^3+6: a. Find d f/d x at x=-5. f'(-5)=

For the function y=f(x)=-2 x^3+6: a. Find d f/d x at x=-5. f'(-5)=
Transcript text: For the function $y=f(x)=-2 x^{3}+6$ : a. Find $\frac{d f}{d x}$ at $x=-5$. \[ f^{\prime}(-5)= \]
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Solution

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Solution Steps

To solve this problem, we need to find the derivative of the function \( f(x) = -2x^3 + 6 \) and then evaluate it at \( x = -5 \).

  1. Compute the derivative of the function \( f(x) \).
  2. Substitute \( x = -5 \) into the derivative to find \( f'(-5) \).
Step 1: Define the Function

Given the function: \[ f(x) = -2x^3 + 6 \]

Step 2: Compute the Derivative

To find the derivative of \( f(x) \), we use the power rule: \[ f'(x) = \frac{d}{dx}(-2x^3 + 6) = -6x^2 \]

Step 3: Evaluate the Derivative at \( x = -5 \)

Substitute \( x = -5 \) into the derivative: \[ f'(-5) = -6(-5)^2 = -6 \cdot 25 = -150 \]

Final Answer

\(\boxed{f'(-5) = -150}\)

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