Question: "How do you use the definition of a..."

High School
Calculus and Analysis
Calculus 1
Derivatives

delitzo1d4

Answered question

2022-04-09

How do you use the definition of a derivative to find the derivative of

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

Answer & Explanation

betazpvaf4

Beginner2022-04-10Added 9 answers

We use our definition as:
$f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$
where h is a small increment.
In our case:
$f^{'} (x) = lim_{h \to 0} \frac{(\frac{1}{x + h} - 2) - (\frac{1}{x} - 2)}{h} =$
$= \lim_{h \to 0} \frac{(\frac{1}{x + h} - 2 - \frac{1}{x) + 2}}{h} =$
$lim_{h \to 0} (\frac{1}{h} (\frac{x - x - h}{x (x + h)})) = lim_{h \to 0} (\frac{1}{h}) (- \frac{h}{x (x + h)}) = lim_{h \to 0} (- \frac{1}{x (x + h)})$
as $h \to 0$
$= (- \frac{1}{x (x + h)}) = - \frac{1}{x^{2}}$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function
$f (x, y) = x^{3} - 6 x y + 8 y^{3}$
$\frac{1}{\sec (x)}$ in derivative?
What is the derivative of $\ln (x + 1)$ ?
What is the limit of $e^{- x}$ as $x \to \infty$ ?
What is the derivative of $f (x) = 5^{\ln x}$ ?
What is the derivative of $e^{- 2 x}$ ?
How to find $lim \frac{e^{t} - 1}{t}$ as $t \to 0$ using l'Hospital's Rule?
What is the integral of $\sqrt{9 - x^{2}}$ ?
What is the derivative of $f (x) = \ln [x^{9} {(x + 3)}^{6} {(x^{2} + 7)}^{5}]$ ?
What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?
How to find the derivative of $y = e^{5 x}$ ?
How to evaluate the limit $\frac{\sin (5 x)}{x}$ as x approaches 0?
How to find derivatives of parametric functions?
What is the antiderivative of $- 5 e^{x - 1}$ ?
How to evaluate: indefinite integral $\frac{1 + x}{1 + x^{2}} d x$ ?

Didn't find what you were looking for?