Rules for Differentiation

The Product Rule

What Is the Product Rule?

The product rule tells you how to find the derivative of the product of two functions. It states:

For $y = uv$ the derivative is given by

y^{'} = u^{'} v + v^{'} u

If $y = x sin (x)$ the derivative is given by

y^{'} = (x sin (x))^{'} = (x)^{'} sin (x) + x (sin (x))^{'}

Using that $(x)^{'} = 1$ and $(sin (x))^{'} = cos (x)$ , you get

y^{'} = sin (x) + x cos (x)

Done