The derivative of the function

  To find the derivative of the function ( f(x) = x^3 + 6x^2 + 9x + 15 ), we can differentiate each term of the function with respect to ( x ) using the power rule of differentiation. Given function: ( f(x) = x^3 + 6x^2 + 9x + 15 ) Taking the derivative of each term: 1. ( \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2 ) 2. ( \frac{d}{dx}(6x^2) = 6 \cdot 2x^{2-1} = 12x ) 3. ( \frac{d}{dx}(9x) = 9 ) 4. ( \frac{d}{dx}(15) = 0 ) (as the derivative of a constant is zero) Therefore, the derivative of the function ( f(x) = x^3 + 6x^2 + 9x + 15 ) is: [ f'(x) = 3x^2 + 12x + 9 ] So, the derived function is ( f'(x) = 3x^2 + 12x + 9 ).    

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