Beginning Activity Beginning Activity 2: Derivatives
In calculus, we learned how to find the derivatives of certain functions. For example, if \(f( x ) = x^2( {\sin x} )\text{,}\) then we can use the product rule to obtain
\begin{equation*}
f'( x ) = 2x( {\sin x} ) + x^2( {\cos x} )\text{.}
\end{equation*}
1.
If possible, find the derivative of each of the following functions:
(a)
\(f( x ) = x^4 - 5x^3 + 3x - 7\)
(b)
\(g( x ) = \cos ( {5x} )\)
(c)
\(h( x ) = \dfrac{{\sin x}}{x}\)
(d)
\(k( x ) = e^{ - x^2 }\)
(e)
\(r( x ) = \left| x \right|\)
2.
Is it possible to think of differentiation as a function? Explain. If so, what would be the domain of the function, what could be the codomain of the function, and what is the rule for computing the element of the codomain (output) that is associated with a given element of the domain (input)?