Answer
a. $f'(a) = -\frac{1}{2} a^{-\frac{3}{2}}$
b. $x+2y = 3$
$x + 16y = 12$
Work Step by Step
$\lim\limits_{h \to 0} \frac{\frac{1}{\sqrt {a+h}}- \frac{1}{\sqrt a}}{h} = $
$\lim\limits_{h \to 0} \frac{\frac{\sqrt a - \sqrt {a+h}}{\sqrt a(\sqrt {a+h})}}{h} = $
$\lim\limits_{h \to 0} \frac{\frac{(\sqrt a - \sqrt {a+h}) (\sqrt a + \sqrt {a+h})}{\sqrt a(\sqrt {a+h}) (\sqrt a + \sqrt {a+h})}}{h} =$
$\lim\limits_{a \to b} \frac{a -(a+h)}{h\sqrt a (\sqrt {a+h}) (\sqrt a + \sqrt {a+h})} = $
Replace $h$ for $0$;
$\lim\limits_{h \to 0} \frac{a-(a+0)}{0\sqrt a (\sqrt {a+0}) (\sqrt a + \sqrt {a+0})} = $
$\frac{-1}{\sqrt a (\sqrt a) (\sqrt a + \sqrt a)} = \frac{-1}{2a \sqrt a}= \frac{-1}{2a^{\frac{3}{2}}} = \frac{-1}{2} a^{-\frac{3}{2}}$
b. Find equations of the tangent lines at the points $(1,1)$ and $(4, \frac{1}{2})$
From part $a$ we know that $f'(a) = -\frac{1}{2} (a)^{-\frac{3}{2}}$
$f(4) = -\frac{1}{2} (4)^{-\frac{3}{2}}$
$f(4) = -\frac{1}{2} * \frac{1}{8}$
$f(4) = -\frac{1}{16}$
$16(y - \frac{1}{2}) = - (x-4)$
$16y - 8 = -x + 4$ Simplify:
$x + 16y = 12$
$f(a) = -\frac{1}{2} (1)^{-\frac{3}{2}}$
$f(1) = -\frac{1}{2} * (1) = -\frac{1}{2}$
$2(y-1)= -(x-1)$
$2y-2 = -x + 1$
$x+2y = 3$
c. To make the graph of the curve and the tangents we graph the curve.
$ y = \frac{1}{\sqrt x}$
And both tangent lines: $x+2y = 3$ and $x + 16y = 12$
