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
pav-90 [236]
3 years ago
9

Use the fundamental definition of a derivative to find f'(x) where f(x)=

%2Bb%7D" id="TexFormula1" title="\frac{x+a}{x+b}" alt="\frac{x+a}{x+b}" align="absmiddle" class="latex-formula">
The answer I get is \frac{-a+b}{\left(x+b\right)^{2}} , but I'm not entirely sure if this is correct and also I'm not sure if I'm using the right method.

Mathematics
2 answers:
IRINA_888 [86]3 years ago
8 0

Answer:

Yes, you are right.

See explanation.

Step-by-step explanation:

The definition of derivative is:

f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}.

We are given f(x)=\frac{x+a}{x+b}.

Assume a \text{ and } b are constants.

If f(x)=\frac{x+a}{x+b} then f(x+h)=\frac{(x+h)+a}{(x+h)+b}.

Let's plug them into our definition above:

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

f'(x)=\lim_{h \rightarrow 0} \frac{\frac{(x+h)+a}{(x+h)+b}-\frac{x+a}{x+b}}{h}

I'm going to find a common denominator for the main fraction's numerator.

That is, I'm going to multiply first fraction by 1=\frac{x+b}{x+b} and

I'm going to multiply second fraction by 1=\frac{(x+h)+b}{(x+h)+b}.

This gives me:

f'(x)=\lim_{h \rightarrow 0} \frac{\frac{((x+h)+a)(x+b)}{((x+h)+b)(x+b)}-\frac{(x+a)((x+h)+b)}{(x+b)((x+h)+b)}}{h}

Now we can combine the fractions in the numerator:

f'(x)=\lim_{h \rightarrow 0} \frac{\frac{((x+h)+a)(x+b)-(x+a)((x+h)+b)}{((x+h)+b)(x+b)}}{h}

I'm going to multiply a bit on top and see if there is anything than can be canceled:

f'(x)=\lim_{h \rightarrow 0} \frac{\frac{(x+h)x+(x+h)b+ax+ab-x(x+h)-xb-a(x+h)-ab}{((x+h)+b)(x+b)}}{h}

Note: I do see that (x+h)x-x(x+h)=0.

I also see ab-ab=0.

I will also distributive in other places in the mini-fraction's numerator.

f'(x)=\lim_{h \rightarrow 0} \frac{\frac{xb+bh+ax-xb-ax-ah}{((x+h)+b)(x+b)}}{h}

Note: I see xb-xb=0.

I also see ax-ax=0.

f'(x)=\lim_{h \rightarrow 0} \frac{\frac{bh-ah}{((x+h)+b)(x+b)}}{h}

In the numerator of the mini-fraction on top the two terms contain a factor of h so I can factor that out.

This will give me something to cancel out across the main fraction since \frac{h}{h}=1.

f'(x)=\lim_{h \rightarrow 0} \frac{\frac{h(b-a)}{((x+h)+b)(x+b)}}{h}

f'(x)=\lim_{h \rightarrow 0} \frac{\frac{(b-a)}{((x+h)+b)(x+b)}}{1}

So we now have gotten rid of what would make this over 0 if we had replace h with 0.

So now to evaluate the limit, that is also we have to do now.

\frac{\frac{(b-a)}{((x+0)+b)(x+b)}}{1}

\frac{\frac{b-a)}{((x)+b)(x+b)}}{1}

\frac{\frac{(b-a)}{(x+b)(x+b)}}{1}

I'm going to go ahead and rewrite this so that isn't over 1 anymore because we don't need the division over 1.

\frac{b-a}{(x+b)(x+b)}

\frac{b-a}{(x+b)^2}

or what you wrote:

\frac{-a+b}{(x+b)^2}

xz_007 [3.2K]3 years ago
4 0

Answer:

(b-a)/(x+b)²

Step-by-step explanation:

f(x+h) = (x+h+a)/(x+h+b)

limit h -->0

[f(x+h) - f(x)]/h

(x+h+a)/(x+h+b) - (x+a)/(x+b)

[(x+b)(x+h+a) - (x+h+b)(x+a)] ÷ [h(x+h+b)(x+b)]

[x²+xb+hx+hb+ax+ab-x²-ax-hx-ha-bx-ab] ÷ [h(x+h+b)(x+b)]

[bh - ah] ÷ [h(x+h+b)(x+b)]

h(b-a) ÷ [h(x+h+b)(x+b)]

(b-a) ÷ [(x+h+b)(x+b)]

As h --> 0

(b-a)/(x+b)²

You might be interested in
A balloon rises to 100 feet in 4 minutes and 125 feet in 5 minutes. Write and solve an equation to find the distance the balloon
Margaret [11]

Answer:

A proportion is the name we give to a statement in which 2 ratios are equal. Proportions are built off of ratios. It is usually written in one of 2 ways

Step-by-step explanation:

7 0
2 years ago
Read 2 more answers
Which function corresponds to the provided graph?
anygoal [31]

Answer:

THE ANSWER is ur mom

Step-by-step explanation:

lol im REPORTING YOU :)

5 0
2 years ago
Solve the following system of equations. <br> 2x-5y=6<br> -2x+10y=-16<br> x=<br> y=
koban [17]
<span>x =<span>−<span><span>2<span> and </span></span>y </span></span></span>=<span>−<span>2. Hope this helps</span></span>
4 0
3 years ago
Find the equation of the line<br> Use exact numbers <br> y=*blank*x+*blank*
Murljashka [212]
You are looking for the slope and y intercept to complete the equation of the line.

The equation of a line is in something called slope intercept form. That looks like y = mx + b. m represents the slope (measure of how steep a line is, and in which direction it is going) and b represents the y intercept (y coordinate when x = 0). You need to find the slope and y intercept to complete the equation.

First, find the slope. The formula for slope is: m = (y2 - y1)/(x2 - x1) where m is the slope and (x1, y1) and (x2, y2) are points.

Pick any two points on the graph. I will use (-2, 0) and (0, 4). Now use these values to find the slope.

m = (4-0)/(0+2) = 4/2
m = 2

m = 2 means that for every two units the line goes up on the y axis, it moves one to the right on the x axis. 2 will go in your first box.

Now find the y intercept. The y intercept is where the line crosses the y axis - it is the y coordinate when x = 0. Here when x = 0, y = 4, so your y intercept is at 4. 4 goes into your second box.

The equation is y = 2x + 4

6 0
2 years ago
What is 14.876 - 4.96
d1i1m1o1n [39]

Answer:

9.916 is the answer

4 0
3 years ago
Read 2 more answers
Other questions:
  • What is the slope of (-1,4)
    15·1 answer
  • Almost Pinejuice (y) costs one quarter as much as pineapple juice (p) does. What equation represents this statement?
    7·2 answers
  • Suppose 20% of the students on campus smoke. you select two students at random. in what percentage of samples will both students
    5·1 answer
  • Pat earns $489 per week at her new job. Calculate her annual salary. (Do not enter $ or comma in your answer.)​
    14·1 answer
  • The line plot shows what fraction of a mile 13 students ran during their fitness test. What is the sum of the shortest distance
    7·2 answers
  • What is the value of this expression when c=4 c^3/2
    5·2 answers
  • Write the equation of the line that passes through the points (4, -3) and
    9·1 answer
  • What is numbers line x&lt;8
    12·1 answer
  • The blue figure is a translation of the black image. Write a rule to describe the translation.
    5·1 answer
  • Taylor states that 2,800,000 in scientific notation is 2.8 x 10 -6 because the number has six places to the right of the 2. Is T
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!