If $$y = ( tan^{-1}x )^2$$, show that $$(x^2 + 1 )^2 \: y_2 + 2x (x^2 +1 ) \: y_1 = 2$$

Toolbox:
• $y=f(x)$
• $\large\frac{dy}{dx}$$=f'(x) • \large\frac{d^2y}{dx^2}=\frac{d}{dx}\big(\frac{dy}{dx}\big) • \large\frac{d}{dx}$$(\tan^{-1}x)=\large\frac{1}{1+x^2}$
Step 1:
We have $y=(\tan^{-1}x)^2$
$y=(\tan^{-1}x)^2$
$y_1=2\tan{-1}x\large\frac{d}{dx}$$(\tan^{-1}x)$
$y_1=2\tan^{-1}x.\large\frac{1}{1+x^2}$
$y_1=\large\frac{2\tan^{-1}x}{1+x^2}$
$(1+x^2)y_1=2\tan^{-1}x$
Step 2:
Differentiating with respect to $x$ we have
$(1+x^2)y_2+y_1(2x)=\large\frac{2}{1+x^2}$
$(1+x^2)^2y_2+2x(1+x^2)y_1=2$
Hence proved.
The passage contains a mathematical problem that requires the reader to prove a given equation based on a function. The solution is provided step-by-step, showing the differentiation process and verifying the equation.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
If $$y = ( tan^{-1}x )^2$$, show that $$(x^2 + 1 )^2 \: y_2 + 2x (x^2 +1 ) \: y_1 = 2$$

Toolbox:
• $y=f(x)$
• $\large\frac{dy}{dx}$$=f'(x) • \large\frac{d^2y}{dx^2}=\frac{d}{dx}\big(\frac{dy}{dx}\big) • \large\frac{d}{dx}$$(\tan^{-1}x)=\large\frac{1}{1+x^2}$
Step 1:
We have $y=(\tan^{-1}x)^2$
$y=(\tan^{-1}x)^2$
$y_1=2\tan{-1}x\large\frac{d}{dx}$$(\tan^{-1}x)$
$y_1=2\tan^{-1}x.\large\frac{1}{1+x^2}$
$y_1=\large\frac{2\tan^{-1}x}{1+x^2}$
$(1+x^2)y_1=2\tan^{-1}x$
Step 2:
Differentiating with respect to $x$ we have
$(1+x^2)y_2+y_1(2x)=\large\frac{2}{1+x^2}$
$(1+x^2)^2y_2+2x(1+x^2)y_1=2$
Hence proved.
user
The passage contains a mathematical problem that requires the reader to prove a given equation based on a function. The solution is provided step-by-step, showing the differentiation process and verifying the equation.

TYPE: SINGLE_QUESTION_WITH_ANSWER