Find the derivative of each of the following functions using the product rule

Question

a. f(x) =x sin x

Answer 1

To find the derivative of f(x) = x sin x using the product rule, we differentiate each function and apply the product rule.

Let u = x and v = sin x.

Using the product rule, the derivative of f(x) is given by:
f'(x) = u'v + uv'

Differentiating u = x, we have:
u' = 1

Differentiating v = sin x, we have:
v' = cos x

Substituting these values into the product rule formula, we get:
f'(x) = (1)(sin x) + (x)(cos x)
f'(x) = sin x + x cos x

Therefore, the derivative of f(x) = x sin x is f'(x) = sin x + x cos x.