Answer:
[tex]k^{\prime}(9) = 1[/tex].
Step-by-step explanation:
The expression [tex]k(x) = x[/tex] is equivalent to [tex]k(x) = x^{1}[/tex].
Apply the power rule of differentiation. For any constant [tex]a[/tex]:
[tex]\displaystyle \frac{d}{dx}[x^{a}] = a\, x^{a-1}[/tex].
In [tex]k(x) = x^{1}[/tex], [tex]a = 1[/tex]. Thus:
[tex]\begin{aligned}k^{\prime}(x) &= \frac{d}{dx}[x^{1}] \\ &= 1\, x^{1 - 1} \\ &= x^{0} \\ &= 1 && \text{given that $x \ne 0$}\end{aligned}[/tex].
In other words, the instantaneous rate of change of [tex]k(x)[/tex] (with respect to [tex]x[/tex]) is constantly [tex]1[/tex] at all [tex]x \ne 0[/tex].
Therefore, for [tex]x = 9[/tex], instantaneous rate of change of [tex]k(x)[/tex] (with respect to [tex]x[/tex]) would be [tex]1[/tex].
