Question

example function of product rule

Answer 1

One example of a function that uses the product rule is the function f(x) = x^2 * sin(x).

To find the derivative of this function, we can apply the product rule.

Let u(x) = x^2 and v(x) = sin(x).

According to the product rule, the derivative of f(x) can be calculated as:

f'(x) = u'(x) * v(x) + u(x) * v'(x)

Taking the derivatives of u(x) and v(x):

u'(x) = 2x (the derivative of x^2)
v'(x) = cos(x) (the derivative of sin(x))

Substituting these values back into the product rule formula:

f'(x) = (2x * sin(x)) + (x^2 * cos(x))

Thus, the derivative of the function f(x) = x^2 * sin(x) is f'(x) = (2x * sin(x)) + (x^2 * cos(x)).