Question

Use the product rule to find the derivative of f(x) = x³cos(x).

Use the product rule to find the derivative of f(x) = (x²+3)(x²-4x).

Answer 1

The derivative of f(x) = x³cos(x) and f(x) = (x²+3)(x²-4x) can be found using the product rule, which results in f'(x) = 3x²cos(x) - x³sin(x) for the first function and f'(x) = 4x³ - 12x² + 6x - 12 for the second function.

Step-by-step explanation:

To find the derivative of f(x) = x³cos(x) and f(x) = (x²+3)(x²-4x), we apply the product rule. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

For the function f(x) = x³cos(x):

1. Let u = x³ and v = cos(x).
2. The derivative of u with respect to x is u' = 3x².
3. The derivative of v with respect to x is v' = -sin(x).
4. Apply the product rule: f'(x) = u'v + uv' = (3x²)cos(x) + x³(-sin(x)) = 3x²cos(x) - x³sin(x).

For the function f(x) = (x²+3)(x²-4x):

1. Let u = x²+3 and v = x²-4x.
2. The derivative of u with respect to x is u' = 2x.
3. The derivative of v with respect to x is v' = 2x - 4.
4. Apply the product rule: f'(x) = u'v + uv' = (2x)(x²-4x) + (x²+3)(2x-4) = 2x³ - 8x² + 2x³ + 6x - 4x² - 12.

After simplification, f'(x) = 4x³ - 12x² + 6x - 12.