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For the following equation, find 
f^(')(x).

f(x)=-2x^(2)+6x+3
Answer: 
f^(')(x)=

For the following equation, find f(x) f^{\prime}(x) .\newlinef(x)=2x2+6x+3 f(x)=-2 x^{2}+6 x+3 \newlineAnswer: f(x)= f^{\prime}(x)=

Full solution

Q. For the following equation, find f(x) f^{\prime}(x) .\newlinef(x)=2x2+6x+3 f(x)=-2 x^{2}+6 x+3 \newlineAnswer: f(x)= f^{\prime}(x)=
  1. Apply Power Rule: To find the derivative of the function f(x)=2x2+6x+3f(x) = -2x^2 + 6x + 3, we will use the power rule for differentiation. The power rule states that the derivative of xnx^n with respect to xx is nx(n1)n\cdot x^{(n-1)}.
  2. Differentiate 2x2-2x^2: Applying the power rule to the first term 2x2-2x^2, we differentiate it as follows:\newlineThe derivative of 2x2-2x^2 with respect to xx is 2×2x21=4x-2 \times 2x^{2-1} = -4x.
  3. Differentiate 6x6x: Next, we apply the power rule to the second term 6x6x, which is a first-degree polynomial. The derivative of 6x6x with respect to xx is 6×1x(11)=66 \times 1x^{(1-1)} = 6.
  4. Differentiate constant: The third term is a constant, 33. The derivative of a constant with respect to xx is 00.
  5. Combine derivatives: Now, we combine the derivatives of all three terms to find the derivative of the entire function f(x)f(x):f(x)=4x+6+0.f'(x) = -4x + 6 + 0.
  6. Simplify final answer: Simplify the derivative to get the final answer: f(x)=4x+6f'(x) = -4x + 6.

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