How do you sketch the graph #y=x^2/(1+x^2)# using the first and second derivatives?
Please see below.
Now for second derivative using quotient formula, it is
The graph appears as follows:
graph{x^2/(1+x^2) [5, 5, 1.62, 3.38]}
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To sketch the graph of ( y = \frac{x^2}{1 + x^2} ) using the first and second derivatives:
 Find the first derivative ( y' ) of the function.
 Find the second derivative ( y'' ) of the function.
 Determine the critical points by setting ( y' = 0 ).
 Test the critical points to determine the nature of the stationary points (whether they are local maximum, local minimum, or inflection points).
 Analyze the behavior of the function as ( x ) approaches positive and negative infinity.
 Sketch the graph based on the above information.
Here are the steps in detail:

Find the first derivative ( y' ) of the function: [ y = \frac{x^2}{1 + x^2} ] [ y' = \frac{(1 + x^2)(2x)  (x^2)(2x)}{(1 + x^2)^2} ] [ y' = \frac{2x(1 + x^2  x^2)}{(1 + x^2)^2} ] [ y' = \frac{2x}{(1 + x^2)^2} ]

Find the second derivative ( y'' ) of the function: [ y'' = \frac{d}{dx} \left( \frac{2x}{(1 + x^2)^2} \right) ] [ y'' = \frac{(1 + x^2)^2(2)  2x(2)(1 + x^2)(2x)}{(1 + x^2)^4} ] [ y'' = \frac{2(1 + x^2  4x^2)}{(1 + x^2)^3} ] [ y'' = \frac{2(1  3x^2)}{(1 + x^2)^3} ]

Determine the critical points by setting ( y' = 0 ): [ \frac{2x}{(1 + x^2)^2} = 0 ] [ 2x = 0 ] [ x = 0 ]

Test the critical point ( x = 0 ): [ y''(0) = \frac{2(1  3 \cdot 0^2)}{(1 + 0^2)^3} = \frac{2}{1} = 2 ] Since ( y''(0) > 0 ), it implies that there's a local minimum at ( x = 0 ).

Analyze the behavior of the function as ( x ) approaches positive and negative infinity: [ \lim_{x \to \infty} y = \lim_{x \to \infty} y = 1 ]

Sketch the graph:
 The function has a local minimum at ( x = 0 ).
 As ( x ) approaches positive and negative infinity, ( y ) approaches 1.
 The graph will be a curve starting from ( y = 1 ) as ( x ) approaches both positive and negative infinity, and reaching its local minimum at ( x = 0 ).
 The curve will be symmetric about the yaxis.
Therefore, sketch the graph accordingly.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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