The Product Rule is a way of differentiating complex expressions that are the product of 2 simpler expressions. This can be expressed algebraically as such:
\(h'(x) = [\textcolor{red}{f(x)}\textcolor{blue}{g(x)}]' = \textcolor{red}{f'(x)}\textcolor{blue}{g(x)} + \textcolor{blue}{g'(x)}\textcolor{red}{f(x)}\)
In short, Product Rule, takes the derivative of \(f(x)\) and multiplies it by \(g(x)\). It then adds this value to \(f(x)\) multiplied by the derivative of \(g(x)\).
It's important to note that using the Product Rule isn't the same as multiplying the derivatives of the respective expressions.
Differentiate \(h(x) = (3x + 2)(x-4)\).
First, we can identify \(f(x)\) and \(g(x)\) and their respective derivatives:
\(\textcolor{red}{f(x) = 3x + 2}\)
\(\textcolor{red}{f'(x) = 3}\)
\(\textcolor{blue}{g(x) = x - 4}\)
\(\textcolor{blue}{g'(x) = 1}\)
Next, we can differentiate the entire expression:
\(h'(x) = \textcolor{red}{f'(x)}\textcolor{blue}{g(x)} + \textcolor{blue}{g'(x)}\textcolor{red}{f(x)}\)
\(h'(x) = \textcolor{red}{3}(\textcolor{blue}{x - 4}) + \textcolor{blue}{1}(\textcolor{red}{3x + 2})\)
\(h'(x) = 3x-12 + 3x + 2\)
\(h'(x) = 6x - 10\)
Therefore, we can determine that \(\boldsymbol{h'(x) = 6x - 10}\).
First, we can identify \(f(x)\) and \(g(x)\) and their respective derivatives:
\(\textcolor{red}{f(x) = x^2+1}\)
\(\textcolor{red}{f'(x) = 2x}\)
\(\textcolor{blue}{g(x) = 1-x}\)
\(\textcolor{blue}{g'(x) = -1}\)
Next, we can differentiate the entire function:
\(p'(x) = \textcolor{red}{f'(x)}\textcolor{blue}{g(x)} + \textcolor{blue}{g'(x)}\textcolor{red}{f(x)}\)
\(p'(x) = \textcolor{red}{2x}(\textcolor{blue}{1-x}) \textcolor{blue}{-1}(\textcolor{red}{x^2+1})\)
\(p'(x) = 2x-2x^2-x^2-1\)
\(p'(x) = x^2+2x-1\)
Therefore, we can determine that \(\boldsymbol{p'(x) = x^2+2x-1}\).