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boyakko [2]
3 years ago
11

What is the value of d/dx(1/x) at x=6 ?​

Mathematics
1 answer:
sladkih [1.3K]3 years ago
4 0

If you know your derivative rules, then

d/d<em>x</em> [1/<em>x</em>] = -1/<em>x</em> ²

so that when <em>x</em> = 6, the derivative has a value of -1/36.

If you have to use the definition of the derivative, then

d/d<em>x</em> [1/<em>x</em>] = lim {<em>h</em> → 0} (1/(<em>x</em> + <em>h</em>) - 1/<em>x</em>) / <em>h</em>

… = lim {<em>h</em> → 0} (<em>x</em> - (<em>x</em> + <em>h</em>)) / (<em>hx</em> (<em>x</em> + <em>h</em>))

… = lim {<em>h</em> → 0} (-<em>h</em>) / (<em>hx</em> (<em>x</em> + <em>h</em>))

… = lim {<em>h</em> → 0} (-1) / (<em>x</em> (<em>x</em> + <em>h</em>))

… = -1/<em>x</em> ²

and at <em>x</em> = 6, you again get -1/36.

Alternatively, use the definition of the derivative at a point:

d/d<em>x</em> [1/<em>x</em>] (6) = lim {<em>x</em> → 6} (1/<em>x</em> - 1/6) / (<em>x</em> - 6)

… = lim {<em>x</em> → 6} ((6 - <em>x</em>) / (6<em>x</em>)) / (<em>x</em> - 6)

… = lim {<em>x</em> → 6} -(<em>x</em> - 6) / (6<em>x</em> (<em>x</em> - 6))

… = lim {<em>x</em> → 6} (-1) / (6<em>x</em>)

… = -1/36

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