Problem

\( f^{\prime}(2), f^{\prime}(x) \) ج \( f(x)=\left(x^{3}+3 x^{2}-3\right)^{\frac{3}{2}} \)

Answer

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Answer

\(f'(2) = 36\sqrt{17}\)

Steps

Step 1 ：\(f'(x) = \frac{3}{2}(x^3 + 3x^2 - 3)^{\frac{1}{2}} \times (3x^2 + 6x)\)

Step 2 ：\(f'(2) = \frac{3}{2}(2^3 + 3(2)^2 - 3)^{\frac{1}{2}} \times (3(2)^2 + 6(2))\)

Step 3 ：\(f'(2) = \frac{3}{2}(8 + 12 - 3)^{\frac{1}{2}} \times (12 + 12)\)

Step 4 ：\(f'(2) = \frac{3}{2}(17)^{\frac{1}{2}} \times 24\)

Step 5 ：\(f'(2) = 36\sqrt{17}\)

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