1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
AVprozaik [17]
2 years ago
14

A) Use the limit definition of derivatives to find f’(x)

Mathematics
1 answer:
Ann [662]2 years ago
8 0
<h3>1)</h3>

\text{Given that,}\\\\f(x) =  \dfrac{ 1}{3x-2}\\\\\text{First principle of derivatives,}\\\\f'(x) = \lim \limits_{h \to 0} \dfrac{f(x+h) - f(x) }{ h}\\\\\\~~~~~~~~= \lim \limits_{h \to 0} \dfrac{\tfrac{1}{3(x+h) - 2} - \tfrac 1{3x-2}}{h}\\\\\\~~~~~~~~= \lim \limits_{h \to 0}  \dfrac{\tfrac{1}{3x+3h -2} - \tfrac{1}{3x-2}}{h}\\\\\\~~~~~~~~= \lim \limits_{h \to 0} \dfrac{\tfrac{3x-2-3x-3h+2}{(3x+3h-2)(3x-2)}}{h}\\\\\\

       ~~~~~~~= \lim \limits_{h \to 0} \dfrac{\tfrac{-3h}{(3x+3h-2)(3x-2)}}{h}\\\\\\~~~~~~~~= \lim \limits_{h \to 0} \dfrac{-3h}{h(3x+3h-2)(3x-2)}\\\\\\~~~~~~~~=-3 \lim \limits_{h \to 0} \dfrac{1}{(3x+3h-2)(3x-2)}\\\\\\~~~~~~~~=-3 \cdot \dfrac{1}{(3x+0-2)(3x-2)}\\\\\\~~~~~~~~=-\dfrac{3}{(3x-2)(3x-2)}\\\\\\~~~~~~~=-\dfrac{3}{(3x-2)^2}

<h3>2)</h3>

\text{Given that,}~\\\\f(x) = \dfrac{1}{3x-2}\\\\\textbf{Power rule:}\\\\\dfrac{d}{dx}(x^n) = nx^{n-1}\\\\\textbf{Chain rule:}\\\\\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}\\\\\text{Now,}\\\\f'(x) = \dfrac{d}{dx} f(x)\\\\\\~~~~~~~~=\dfrac{d}{dx} \left( \dfrac 1{3x-2} \right)\\\\\\~~~~~~~~=\dfrac{d}{dx} (3x-2)^{-1}\\\\\\~~~~~~~~=-(3x-2)^{-1-1} \cdot \dfrac{d}{dx}(3x-2)\\\\\\~~~~~~~~=-(3x-2)^{-2} \cdot 3\\\\\\~~~~~~~~=-\dfrac{3}{(3x-2)^2}

You might be interested in
Write y=2(x + 6)(x + 3) in standard form
Volgvan

Answer:

y = 2x² + 18x + 36

Step-by-step explanation:

Given

y = 2(x + 6)(x + 3) ← expand the factors using FOIL

y = 2(x² + 9x + 18) ← distribute parenthesis by 2

  = 2x² + 18x + 36 ← in standard form

6 0
3 years ago
39. Kate recorded the time it took six children of
scZoUnD [109]

Answer:

y=-11x+261

Step-by-step explanation:

As you can observe in the image attached, the line that best fits passes through point B and C. That means we can use those point to find the slope of such line.

m=\frac{y_{2} -y_{1} }{x_{2}-x_{1}  }

Where B(11,137) and C(12,126)

m=\frac{126-137}{12-11}=-11

So, the slope of the line that best fits is -11, approximately.

Now, we use the point-slope formula to find the equation.

y-y_{1} =m(x-x_{1} )\\y-137=-11(x-11)\\y=-11x+124+137\\y=-11x +261

Therefore, the line that best fits is y=-11x+261 approximately.

Remember, when we estimate a line for some data on a scatterplot, we are calculating an approximation, that's why we also said "approximately", because the line is an approximation where the majority of point meet.

4 0
3 years ago
Can someone plz solve this for me?
ElenaW [278]

Answer:yellow box


Step-by-step explanation:


7 0
3 years ago
Increase £240 by 30%?
laila [671]


240*0.30=72    

72+240=312 euro

5 0
3 years ago
Read 2 more answers
Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s). 3 to the power
Andreas93 [3]

Answer:

For this expression, a = 3, b = 6, and c = 5.

Step-by-step explanation:

8 0
2 years ago
Other questions:
  • Write the number 2 and a half million in figures
    15·1 answer
  • Don’t understand again?
    9·1 answer
  • Jess owns 4 red shirts, 2 blue shirts, and 3 white shirts. She chooses a shirt to wear at random. What is the probability that s
    10·2 answers
  • Elizabeth quiere organizar la fiesta de su cumpleaños y pretende invitar a 20 de sus amigos. Para los preparativos de la fiesta
    5·1 answer
  • What is/are the classification of functions that has its inverse?
    6·1 answer
  • The Wood frog has three unique chemicals inside its body that act like antifreeze in your car to keep the frog from freezing in
    12·1 answer
  • Find the value of x.
    14·1 answer
  • Suppose that the local sales tax rate is 5 %
    6·1 answer
  • 5
    8·2 answers
  • Find the surface area of a hemisphere that has a volume of 486π <img src="https://tex.z-dn.net/?f=cm%5E%7B3%7D" id="TexFormula1"
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!