Preview Activity 2.7.1.
Let \(f\) be a differentiable function of \(x\) (whose formula is not known) and recall that \(\frac{d}{dx}[f(x)]\) and \(f'(x)\) are interchangeable notations. Determine each of the following derivatives of combinations of explicit functions of \(x\text{,}\) the unknown function \(f\text{,}\) and an arbitrary constant \(c\text{.}\)
(a)
\(\frac{d}{dx} \left[ x^2 + f(x) \right]\)
(b)
\(\frac{d}{dx} \left[ x^2 f(x) \right]\)
(c)
\(\frac{d}{dx} \left[ c + x + f(x)^2 \right]\)
(d)
\(\frac{d}{dx} \left[ f(x^2) \right]\)
(e)
\(\frac{d}{dx} \left[ xf(x) + f(cx) + cf(x) \right]\)
