Explanation
The product rule is a rule used in differentiation when you want to differentiate two functions that are multiplied together. When differentiating products, you need to apply the product rule. The product rule formula is:
Procedure
To differentiate products, you must first identify the two functions u(x) and v(x). Then differentiate each function separately and insert these derivatives into the product rule formula. Afterwards, simplify the expression.
Examples
Differentiate the function f(x) = x \cdot \sin(x).
Let u(x) = x and v(x) = \sin(x). Then we have u'(x) = 1 and v'(x) = \cos(x).
Inserting into the product rule formula gives:
Differentiate the function f(x) = (2x^2 + 3) \cdot e^{x}.
Let u(x) = 2x^2 + 3 and v(x) = e^{x}. Then we have u'(x) = 4x and v'(x) = e^{x}.
Note
- ★The product rule is used to differentiate the product of two functions.
- ★The product rule formula is: (u(x) \cdot v(x))' = u'(x) \cdot v(x) + u(x) \cdot v'(x).
Exercises
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