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

Use limit definition of derivative to find:f′(x)=−1x2Explanation:f′(x)=xlimh→0(f(x+h)−f(x)h)=limh→0(1x+h−1xh)=limh→0(x−(x+h)hx(x+h))=limh→0(−hhx(x+h))=limh→0(−1x(x+h))=−1x2

Try this:limh→01x+h−1xh=limh→01h(x−x−h(x+h)x)=limh→01h(−h(x+h)x)=Simplifying h and taking the limit:−1x2

Use limit definition of derivative to find:f′(x)=−1x2Explanation:f′(x)=xlimh→0(f(x+h)−f(x)h)=limh→0(1x+h−1xh)=limh→0(x−(x+h)hx(x+h))=limh→0(−hhx(x+h))=limh→0(−1x(x+h))=−1x2

Try this:limh→01x+h−1xh=limh→01h(x−x−h(x+h)x)=limh→01h(−h(x+h)x)=Simplifying h and taking the limit:−1x2

"How do you find f'(x) using the definition..." solved and explained

High School
Calculus and Analysis
Calculus 1
Derivatives

Malachi Mullins

Answered question

2022-04-17

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

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

Answer & Explanation

cutimnm135imsa

Beginner2022-04-18Added 21 answers

Use limit definition of derivative to find:

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

Explanation:

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

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

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

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

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

= - \frac{1}{x^{2}}

cinereod3am

Beginner2022-04-19Added 10 answers

Try this:

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

Simplifying h and taking the limit:

- \frac{1}{x^{2}}

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