![]() = lim h → 0 g ( x ) ⋅ lim h → 0 f ( x + h ) - f ( x ) h - lim h → 0 f ( x ) ⋅ lim h → 0 g ( x + h ) - g ( x ) h lim h → 0 g ( x + h ) ⋅ lim h → 0 g ( x ) = lim h → 0 g ( x ) f ( x + h ) - f ( x ) h - f ( x ) g ( x + h ) - g ( x ) h g ( x + h ) g ( x ) It is straightforward to extend this pattern to finding the derivative of a product of 4 or more functions.Īdding and subtracting the term f ( x ) g ( x ) in the numerator does not change the value of the expression and allows us to separate f and g so that Each term contains only one derivative of one of the original functions, and each function’s derivative shows up in only one term. Recognize the pattern in our answer above: when applying the Product Rule to a product of three functions, there are three terms added together in the final derivative. = x 3 ln x ( - sin x ) + x 3 1 x cos x + 3 x 2 ln x cos x = ( x 3 ) ( ln x ( - sin x ) + 1 x cos x ) + 3 x 2 ( ln x cos x ) ![]() To evaluate ( ln x cos x ) ′, we apply the Product Rule again: ![]() = ( x 3 ) ( ln x cos x ) ′ + 3 x 2 ( ln x cos x ) Our method of handling this problem is to simply group the latter two functions together, and consider y = x 3 ( ln x cos x ). ![]() SolutionWe have a product of three functions while the Product Rule only specifies how to handle a product of two functions. ![]()
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