Question

Find the coefficient of the squared term in the simplified form for the second derivative., f"(x) for f(x)=(X^3+2x+3)(3x^3-6x^2-8x+1). use the hyphen symbol,-, for negative values.

( Fill-in-the-blank)

Answer 1

Use the product rule:
f'(x) =(x³+2x+3)'(3x³-6x²-8x+1) + (x³+2x+3)(3x³-6x²-8x+1)'
f'(x) =(3x²+2)(3x³-6x²-8x+1) + (x³+2x+3)(6x²-12x-8)

Use the product rule again:
f''(x) = (6x)(3x³-6x²-8x+1) + (3x²+2)(9x²-12x-8) + (3x²+2)(6x²-12x-8) + (x³+2x+3)(12x-12)

We only care about the coefficient of the x² term so let's extract the operations of terms that give us x²: (6x)(-8x)+(3x²)(-8)+2(9x²)+(3x²)(-8)+2(6x²)+(2x)(12x) = - 48x²+(-24x²)+18x²+(-24x²)+12x²+24x² = 54x²