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When and how do I use the product rule for differentiation?

As the name suggests, the product rule is used to differentiate a function in which a product of 2 expressions in x exists. This means the two expressions in x are multiplied by each other, even when the function is expressed in its simplest form. An example would be y=x3e2x. The product rule is written by generalising one expression in x as u and the other as v: 

If y=u*v then

dy/dx= u*dv/dx + v*du/dx

This means that, to dfferentiate, we multiply each expression in x by the derivative of the other and add the results. This is illustrated by the example below: 

 y=x3e2x

let u= x3                 v=e2x

du/dx = 3x2            dv/dx= 2e2x

for this example: 

                       dy/dx = u dv/dx + v du/dx

                                = x3*2e2x +  e2x*3x2 

                                = e2x(2x3 + 3x2)

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