product rule
The product rule is a formula used in calculus to find the derivative of a function that is the multiplication of two or more functions
The product rule is a formula used in calculus to find the derivative of a function that is the multiplication of two or more functions. It allows us to differentiate the product of two functions without having to use the limit definition of the derivative.
The product rule states:
If u(x) and v(x) are two functions of x, then the derivative of their product, u(x) * v(x), is given by:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
Here, u'(x) represents the derivative of u(x) with respect to x, and v'(x) represents the derivative of v(x) with respect to x.
To apply the product rule, follow these steps:
1. Identify the functions u(x) and v(x) in the given problem.
2. Differentiate each function separately, finding u'(x) and v'(x).
3. Substitute the values of u'(x) and v'(x) into the formula:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x).
Let’s illustrate the product rule with an example:
Example:
Find the derivative of f(x) = x^2 * sin(x).
Here, u(x) = x^2 and v(x) = sin(x).
1. Differentiate each function separately:
u'(x) = 2x (using the power rule)
v'(x) = cos(x) (using the derivative of sin(x))
2. Use the product rule formula:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
(d/dx)(x^2 * sin(x)) = (2x * sin(x)) + (x^2 * cos(x))
Therefore, the derivative of f(x) = x^2 * sin(x) is f'(x) = 2x * sin(x) + x^2 * cos(x).
Note that the product rule can also be extended to the case of three or more functions being multiplied together, with additional terms added in the formula for each additional function.
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