Let \( f(x) = -3x^2(1 - x) - 3x(1 - x)^2 - (1 - x)^3 \). Then, \( \frac{df(x)}{dx} = \)
Let \( f(x) = -3x^2(1 - x) - 3x(1 - x)^2 - (1 - x)^3 \). Then, \( \frac{df(x)}{dx} = \)
Show Hint
- \( 3x^2 \)
- \( 3(1 - x)^2 \)
- \( 3x(1 - x) \)
- \( x \)
The Correct Option is A
Solution and Explanation
\[ f(x) = -3x^2(1 - x) - 3x(1 - x)^2 - (1 - x)^3 \] First, expand each term: \[ -3x^2(1 - x) = -3x^2 + 3x^3 \] \[ -3x(1 - x)^2 = -3x(1 - 2x + x^2) = -3x + 6x^2 -3x^3 \] \[ -(1 - x)^3 = -(1 - 3x + 3x^2 - x^3) = -1 + 3x -3x^2 + x^3 \] Add all the terms: \[ f(x) = (-3x^2 + 3x^3) + (-3x + 6x^2 - 3x^3) + (-1 + 3x -3x^2 + x^3) \] Combine like terms: \[ f(x) = (-3x^2 + 6x^2 -3x^2) + (3x^3 - 3x^3 + x^3) + (-3x + 3x) + (-1) \] \[ = 0x^2 + x^3 - 1 = x^3 - 1 \] Step 2: Differentiate \( f(x) = x^3 - 1 \).
\[ \frac{df(x)}{dx} = 3x^2 \]
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