How do you find f'(x) using the definition of a derivative for f(x)=1x2?

f(x)=1x2f′(x)=limh→0f(x+h)−f(x)h=limh→01(x+h)2−1x2h=limh→0x2−(x+h)2h(x+h)2x2=limh→0−2xh−h2h(x+h)2x2=limh→0−2x−h(x+h)2x2=−2x3Result:f′(x)=−2x3

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Zane Decker

2022-04-15

How do you find f'(x) using the definition of a derivative for

f (x) = \frac{1}{x^{2}}

Wernbergbo9d

Beginner2022-04-16Added 12 answers

f (x) = \frac{1}{x^{2}}

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

= lim_{h \to 0} \frac{\frac{1}{{(x + h)}^{2} - \frac{1}{x^{2}}}}{h}

= lim_{h \to 0} \frac{x^{2} - {(x + h)}^{2}}{h {(x + h)}^{2} x^{2}}

= lim_{h \to 0} \frac{- 2 x h - h^{2}}{h {(x + h)}^{2} x^{2}}

= lim_{h \to 0} \frac{- 2 x - h}{{(x + h)}^{2} x^{2}}

= - \frac{2}{x^{3}}

Result:

f^{'} (x) = - \frac{2}{x^{3}}

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