Product Rule Differentiation , Differentiation Formulas

The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and is given by. . Remember the rule in the following way. Each time, differentiate a different function in the product and add the two terms together.

The product rule is used in calculus when you are asked to take the derivative of a function that is the multiplication of a couple or several smaller functions. In other words, a function f(x) is a product of functions if it can be written as g(x)h(x), and so on.
product rule differentiation

The Power of a Product rule is another way to simplify exponents. First, we need to define some terms as they relate to exponents. When you have a number or variable raised to a power, it is called the base, while the superscript number, or the number after the '^' mark, is called the exponent or power.

product rule for differentiation

Examples. Suppose we want to differentiate ƒ(x) = x2sin(x). By using the product rule, one gets the derivative ƒ '(x) = 2x sin(x) + x2cos(x) (since the derivative of x2 is 2x and the derivative of sin(x) is cos(x)). ... This, combined with the sum rule for derivatives, shows that differentiation is linear.

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