Question

Find the derivative of the function. g(x)=e x 3 −2x g ′ (x)=(e) x 3 −2x (3x−2) ​ If f and g are the functions whose graphs are shown, let u(x)=f(x)g(x) and v(x)= g(x) f(x) ​ .

Answer 1

`\[ u'(x) = g(x) \cdot f'(x) + f(x) \cdot g'(x) \]`

Step-by-step explanation:

The product rule states that the derivative of the product of two functions is given by the first function times the derivative of the second function plus the second function times the derivative of the first function. In the context of the given question, if \( u(x) = f(x) \cdot g(x) \), then the derivative \( u'(x) \) is calculated using the product rule. The term \( g(x) \cdot f'(x) \) corresponds to the first function (g(x)) times the derivative of the second function (f(x)), and the term \( f(x) \cdot g'(x) \) corresponds to the second function (f(x)) times the derivative of the first function (g(x)).

This rule is derived from the limit definition of the derivative. When you have a product of two functions, the change in the product can be expressed as the sum of two parts: the change in the first function times the second function, and the change in the second function times the first function. The product rule is a concise way of expressing this relationship and is a fundamental tool in calculus for finding the derivatives of products of functions. In the given context, applying the product rule to
`\( u(x) = f(x) \cdot g(x) \) results in the final answer \( u'(x) = g(x) \cdot f'(x) + f(x) \cdot g'(x) \).`