# What is the second derivative of $\dfrac{1}{1+{{x}^{2}}}$ ?

Answer

Verified

279.6k+ views

**Hint:**With the help of a second derivative, we see how the rate of change of quantity is itself changing. Let us see an example: If we take the second derivative of the position of the object with respect to time is the instantaneous acceleration of the object or we can say it is the rate at which the velocity of the object is changing with respect to time.

**Complete step-by-step solution:**

We use the second derivative of a function for various purposes such as to check the concavity of the graph. If the second derivative of the function is positive, we say that the graph is concave up, otherwise we say that it is concave down. Concave up is also said to be a convex graph where the tangent line lies below the graph.

We can also use the relation between the second derivative of the function and the graph to check whether a stationary point for a function is a local maximum or a local minimum point.

Now, we are given in the question to find the second derivative of $f(x)=\dfrac{1}{1+{{x}^{2}}}$

For which we will proceed like this:

Now using the chain rule, we will get

$\begin{align}

& f'(x)=\dfrac{d{{(1+{{x}^{2}})}^{-1}}}{dx} \\

& =(2x)(-1){{(1+{{x}^{2}})}^{-2}} \\

& =\dfrac{-2x}{{{(1+{{x}^{2}})}^{2}}} \\

\end{align}$

Now using the quotient rule which is

$\begin{align}

& f'\left( x \right)=\dfrac{u\left( x \right)}{v\left( x \right)} \\

& \Rightarrow f''\left( x \right)=\dfrac{u'\left( x \right)v\left( x \right)-u\left( x \right)v'\left( x \right)}{{{\left( v\left( x \right) \right)}^{2}}} \\

\end{align}$

we will get the second derivative of the given function and the first derivative of the first derivative of given function which is:

$\begin{align}

& u\left( x \right)=-2x \\

& v\left( x \right)={{\left( 1+{{x}^{2}} \right)}^{2}} \\

\end{align}$

Now, we know that derivative of polynomials in the form ${{x}^{n}}$ is given by $n{{x}^{n-1}}$ therefore, derivative of $u(x)=-2$ as n is 1 and n-1 is 0 which is equivalent to 1.

Similarly, derivative of

$\begin{align}

& v\left( x \right)=2\left( 1+{{x}^{2}} \right)\cdot 2x \\

& \Rightarrow 4x\left( 1+{{x}^{2}} \right) \\

\end{align}$

Now, using this we get,

$\begin{align}

&f''(x)=\dfrac{(-2){{(1+{{x}^{2}})}^{2}}+2x\times 2\left( 1+{{x}^{2}} \right)\times 2x}{{{(1+{{x}^{2}})}^{4}}} \\

& \Rightarrow f''\left( x \right)=\dfrac{(-2)(1+{{x}^{2}})+8{{x}^{2}}}{{{(1+{{x}^{2}})}^{3}}} \\

\end{align}$

**Therefore, the second derivative of the given function is $f''\left( x \right)=\dfrac{(-2)(1+{{x}^{2}})+8{{x}^{2}}}{{{(1+{{x}^{2}})}^{3}}}$.**

**Note:**To find the second derivative firstly simply the first derivative and simplify it. We exactly need to know what is derivative and what is the significance of the second derivative and how it helps in different applications.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What are the measures one has to take to prevent contracting class 12 biology CBSE

Suggest some methods to assist infertile couples to class 12 biology CBSE

Amniocentesis for sex determination is banned in our class 12 biology CBSE

Trending doubts

The ray passing through the of the lens is not deviated class 10 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

What is the nlx method How is it useful class 11 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the difference between anaerobic aerobic respiration class 10 biology CBSE