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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=x^{3}e^{2x}. 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=x^{3}e^{2x}

let u= x^{3} v=e^{2x}

du/dx = 3x^{2} dv/dx= 2e^{2x}

for this example:

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

= x^{3}*2e^{2x} + e^{2x}*3x^{2}

= e^{2x}(2x^{3} + 3x^{2})