How do I use the product rule for differentiation?

The product rule is a rule for differentiating products of expressions, i.e. 2 expressions multiplied together. 

The rule is: if y=AB, then dy/dx=AdB/dx +B*dA/dx. This can be remembered as "Write down the first, differentiate the second then write down the second, differentiate the first". 

For example:

Differentiate y= e^2x sin(x) with respect to x.

This is a product rule question because there are 2 expressions multiplied together.

To answer this question, we write down the 2 expressions that have been multiplied together:

A=e^2x and B=sin(x)

Then, we differentiate each of them

dA/dx =2e^2x  (Remember that if you have to differentiate e^ax, it equals a*e^ax)

and

 dB/dx=cos(x) (remember that differentiating sin(x) gives cos(x)).

Next, we combine these expressions using the rule at the top

so dy/dx=e^2x cos(x) + sin(x) *2e^2x

As a final step, we can simplify this by  taking out a factor of e^2x, so

dy/dx=e^2x (cos(x) + 2sin(x)).