As you know, the product rule for taking the derivative is \[ [f(x)g(x)]' = f(x)g'(x) + f'(x)g(x) \] One of the most common mistakes students do when first learning this rule is to take the derivative of each function and then multiply the results, like we do when two functions are added together. The correct equation is based on limits.
When does \( [f(x)g(x)]' = f'(x)g'(x) \) hold?
It would be nice if we could take the derivative of each function and then just multiply them together. But this will not work nearly all the time. This video derives the equation for the one unique time when this equation holds.
Here is another video that may help you better understand this.
Really UNDERSTAND Calculus
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