Question

Compute the derivative of h(x)= 2x²3x(4−x).

Answer 1

To compute the derivative of the function h(x)=2x²3x(4−x), use the product rule. Apply the product rule to calculate the derivatives of each term and simplify the equation.

Step-by-step explanation:

To compute the derivative of the function h(x)=2x²3x(4−x), we can use the product rule. The product rule states that if we have a function of the form f(x) = g(x) * h(x), then the derivative of f(x) is given by f'(x) = g'(x) * h(x) + g(x) * h'(x).

Applying the product rule to h(x), we get:

h'(x) = (2x² * 3x(4−x))' = (2x²)' * 3x(4−x) + 2x² * (3x(4−x))'

Now, we can calculate the derivatives of the terms:

(2x²)' = 4x, and (3x(4−x))' = 3 * (4−x) + 3x * (-1) = 12 - 3x - 3x = 12 - 6x

Substituting these derivatives back into the equation, we get:

h'(x) = 4x * 3x(4−x) + 2x² * (12 - 6x)

Simplifying further gives:

h'(x) = 12x²(4−x) + 24x² - 12x³

Therefore, the derivative of h(x) is h'(x) = 12x²(4−x) + 24x² - 12x³.