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
Vladimir [108]
1 year ago
7

If

Formula1" title="\mathrm {y = (x + \sqrt{1+x^{2}})^{m}}" alt="\mathrm {y = (x + \sqrt{1+x^{2}})^{m}}" align="absmiddle" class="latex-formula">, then prove that \mathrm {(x^{2} +1)y_{2} +x y_{1} - m^{2}y = 0}.
Note : y₁ and y₂ refer to the first and second derivatives.
Mathematics
1 answer:
Harman [31]1 year ago
8 0

Answer:

See below for proof.

Step-by-step explanation:

<u>Given</u>:

y=\left(x+\sqrt{1+x^2}\right)^m

<u>First derivative</u>

\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule for Differentiation}\\\\If  $f(g(x))$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=f'(g(x))\:g'(x)$\\\end{minipage}}

<u />

<u />\boxed{\begin{minipage}{5 cm}\underline{Differentiating $x^n$}\\\\If  $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=xn^{n-1}$\\\end{minipage}}

<u />

\begin{aligned} y_1=\dfrac{\text{d}y}{\text{d}x} & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(1+\dfrac{2x}{2\sqrt{1+x^2}} \right)\\\\ & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(1+\dfrac{x}{\sqrt{1+x^2}} \right) \\\\ & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(\dfrac{x+\sqrt{1+x^2}}{\sqrt{1+x^2}} \right)\\\\ & = \dfrac{m}{\sqrt{1+x^2}} \cdot \left(x+\sqrt{1+x^2}\right)^{m-1}  \cdot \left(x+\sqrt{1+x^2}\right)\\\\ & = \dfrac{m}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m\end{aligned}

<u>Second derivative</u>

<u />

\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If  $y=uv$  then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}

\textsf{Let }u=\dfrac{m}{\sqrt{1+x^2}}

\implies \dfrac{\text{d}u}{\text{d}x}=-\dfrac{mx}{\left(1+x^2\right)^\frac{3}{2}}

\textsf{Let }v=\left(x+\sqrt{1+x^2}\right)^m

\implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{m}{\sqrt{1+x^2}} \cdot \left(x+\sqrt{1+x^2}\right)^m

\begin{aligned}y_2=\dfrac{\text{d}^2y}{\text{d}x^2}&=\dfrac{m}{\sqrt{1+x^2}}\cdot\dfrac{m}{\sqrt{1+x^2}}\cdot\left(x+\sqrt{1+x^2}\right)^m+\left(x+\sqrt{1+x^2}\right)^m\cdot-\dfrac{mx}{\left(1+x^2\right)^\frac{3}{2}}\\\\&=\dfrac{m^2}{1+x^2}\cdot\left(x+\sqrt{1+x^2}\right)^m+\left(x+\sqrt{1+x^2}\right)^m\cdot-\dfrac{mx}{\left(1+x^2\right)\sqrt{1+x^2}}\\\\ &=\left(x+\sqrt{1+x^2}\right)^m\left(\dfrac{m^2}{1+x^2}-\dfrac{mx}{\left(1+x^2\right)\sqrt{1+x^2}}\right)\\\\\end{aligned}

              = \dfrac{\left(x+\sqrt{1+x^2}\right)^m}{1+x^2}\right)\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)

<u>Proof</u>

  (x^2+1)y_2+xy_1-m^2y

= (x^2+1) \dfrac{\left(x+\sqrt{1+x^2}\right)^m}{1+x^2}\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)+\dfrac{mx}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m-m^2\left(x+\sqrt{1+x^2\right)^m

= \left(x+\sqrt{1+x^2}\right)^m\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)+\dfrac{mx}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m-m^2\left(x+\sqrt{1+x^2\right)^m

= \left(x+\sqrt{1+x^2}\right)^m\left[m^2-\dfrac{mx}{\sqrt{1+x^2}}+\dfrac{mx}{\sqrt{1+x^2}}-m^2\right]

= \left(x+\sqrt{1+x^2}\right)^m\left[0]

= 0

You might be interested in
1. What events led to the attack at Pearl Harbor?
umka21 [38]
War between Japan and the United States.
4 0
3 years ago
Read 2 more answers
An octagonal pyramid ... how many faces does it have, how many vertices and how many edges? A triangular prism ... how many face
nadezda [96]

1: 8 faces and 9 with the base 9 vertices and 16 edges

2: 3 faces and 5 with the bases 6 vertices and 9 edges

3: 3 faces and 4 with the base 4 vertices and 6 edges

Hope this can help you.

4 0
3 years ago
Which number forms a Pythagorean Triple with 8 and 10? <br>A: 6 <br>B: 15
daser333 [38]
Are 6 and 15 the legs of the triangle?
6 0
3 years ago
Read 2 more answers
Does anyone know if this is correct or how to do this? (part a,b, and c)
stepladder [879]

 Part A you would use 200-L ( since you have 400 feet total 200 feet would equal length plus width)

Part B 200 - 80 = 120

Part C 200-90 = 110

area = 90 x 110 = 9900 square feet

5 0
3 years ago
I understand the equation, it's just not working out. Can anyone help me figure it out? Thanks in advance.
andreyandreev [35.5K]
You put the 15x + 100 on the angle CEB instead of CED. If you put it in the right spot, it should work out fine since they are vertical angles.
4 0
3 years ago
Other questions:
  • The recipe for Ryan birthday cake calls for 3/4 of a cup of flour and 2/4 of a cup of sugar. How many total cups of flour and su
    6·2 answers
  • The set of integers greater than -3 and less or equal to 5
    9·1 answer
  • Me.Webster is buying carpet for n exercise room in his basement. The room will have an area of 230 square ft. The width of the r
    8·1 answer
  • I need to model 2 equivalent fractions for 6 fifths
    6·1 answer
  • a large storm hit southern california. it rains 0.6 in in 2 hours. after 5 hours, it rained 1.5 in. what if the storm continues
    10·1 answer
  • Gio’s soccer team is having a car wash to raise money for new uniforms. C stands for how many cars are washed and D stands for h
    5·2 answers
  • Find the value of x that will make L parallel to m
    12·2 answers
  • What is the equation of the line?
    12·1 answer
  • Please help me..!!.....
    5·1 answer
  • Pa help huhu need lngs mga lods​
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!