Question1. Use the definition to find the derivative of each of the following functions: (a) $f(x):=x^{3}$ for $x \in \mathbb{R}$, (c) $h(x):=\sqrt{x}$ for $x>0$, (b) $g(x):=1 / x$ for $x \in \mathbb{R}, x \neq 0$, (d) $k(x):=1 / \sqrt{x}$ for $x>0$.

Studdy Solution
As $h \to 0$, the expression simplifies to:
$$k'(x) = -\frac{1}{2x\sqrt{x}}$$
The derivatives of the functions are: (a) $f'(x) = 3x^2$ (b) $g'(x) = -\frac{1}{x^2}$ (c) $h'(x) = \frac{1}{2\sqrt{x}}$ (d) $k'(x) = -\frac{1}{2x\sqrt{x}}$