Product rule for differentiation is different from the sum and difference rule for derivatives. In sum and difference rules the derivative of a sum or difference is the sum or difference of the derivatives. In other words we find the derivatives separate and add or subtract as required.
But the derivative of the product of a function is not the product of derivatives.
For example, consider f(x) = x5. By using the power rule for derivative of a function we get f'(x) = 5x4.  If we treat the function f(x) as product as f(x) = x3.x2, then  the derivative of x3 = 3x2 and the derivative of x2 = 2x. If we multiply the two derivatives we get (3x2).(2x) = 6x3 which is not the derivative of x5 .Hence the differentiation of a product is analyzed using the definition of the derivative and a rule is arrived at for finding the derivative of a function which is expressed as a product of two or more functions.
Many rules exist under the name Product Rule related to various concepts like the product rule for exponents, product rule for logarithms, product rule for limits and product rule for derivatives. Let us first view few instances where the product rules are defined other than for derivatives.

The Product Rule for derivatives can be defined as follows.

If two functions are differentiable, so is their product and the derivative of their product is the sum of the products obtained by multiplying a function with the derivative of the other function.
Using notation we can write this as

$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x)+g(x)f'(x)$