Question

How to find the derivative of \[{{x}^{3}}\arctan \left( 7x \right)\]?

Answer 1

Hint: To find the derivative first apply the product rule in the given function \[{{x}^{3}}\arctan \left( 7x \right)\]and the formula of product rule is: \[{{\left[ u\left( x \right).v\left( x \right) \right]}^{\prime }}=u\left( x \right).{v}'\left( x \right)+v\left( x \right).{u}'\left( x \right)\] where \[u\left( x \right)={{x}^{3}}\]and \[v\left( x \right)=\arctan \left( 7x \right)\]. Secondly, to further differentiate \[{{x}^{3}}\]apply differentiation rule that is \[{u}'\left( x \right)=n{{x}^{n-1}}\] where according to the question \[n\] is equal to \[3\] & \[u\left( x \right)={{x}^{3}}\]and to differentiate \[\arctan \left( 7x \right)\] apply differentiation rule: \[{{\left[ \arctan \left( v\left( x \right) \right) \right]}^{\prime }}=\dfrac{1}{v{{\left( x \right)}^{2}}+1}\cdot {v}'\left( x \right)\] here \[v\left( x \right)=7x\]. Then if possible, simplify the solution.

Complete step by step solution:
The derivative of \[{{x}^{3}}\arctan \left( 7x \right)\] is as follows:
\[\dfrac{d}{dx}\left[ {{x}^{3}}\arctan \left( 7x \right) \right]\]
Applying product rule: \[{{\left[ u\left( x \right).v\left( x \right) \right]}^{\prime }}=u\left( x \right).{v}'\left( x \right)+v\left( x \right).{u}'\left( x \right)\] in the given function we get:
\[\Rightarrow \dfrac{d}{dx}\left[ {{x}^{3}} \right]\cdot \arctan \left( 7x \right)+{{x}^{3}}\cdot \dfrac{d}{dx}\left[ \arctan \left( 7x \right) \right]...(i)\]
Now to further differentiate \[{{x}^{3}}\]apply differentiation rule that is \[{u}'\left( x \right)=n{{x}^{n-1}}\] where according to the question \[n\] is equal to \[3\] & \[u(x)={{x}^{3}}\]that is
\[\Rightarrow \dfrac{d}{dx}\left[ {{x}^{3}} \right]=3{{x}^{2}}...(ii)\]
and to differentiate \[\arctan (7x)\] apply differentiation rule: \[{{\left[ \arctan \left( v\left( x \right) \right) \right]}^{\prime }}=\dfrac{1}{v{{\left( x \right)}^{2}}+1}\cdot {v}'\left( x \right)\] where \[v\left( x \right)=7x\] that is
\[\Rightarrow \dfrac{d}{dx}\left[ \arctan \left( 7x \right) \right]=\dfrac{1}{{{\left( 7x \right)}^{2}}+1}\cdot \dfrac{d}{dx}\left[ 7x \right]...(iii)\]
Now putting the values of equation \[(ii)\]and \[(iii)\] in equation \[(i)\] we get:
\[\Rightarrow 3{{x}^{2}}\cdot \arctan \left( 7x \right)+{{x}^{3}}\cdot \dfrac{1}{{{\left( 7x \right)}^{2}}+1}\cdot \dfrac{d}{dx}\left[ 7x \right]...(iv)\]
According to the differentiation rule that is \[{u}'\left( x \right)=n{{x}^{n-1}}\] we know that derivative of \[7x\] is
\[\Rightarrow \dfrac{d}{dx}\left[ 7x \right]=7...(v)\]
Now putting the value of equation \[(v)\] in equation \[(iv)\] and multiplying the terms we get:
\[\Rightarrow 3{{x}^{2}}\cdot \arctan \left( 7x \right)+\dfrac{{{x}^{3}}\cdot 7\cdot 1}{{{\left( 7x \right)}^{2}}+1}\]
We know that \[{{7}^{2}}\] is equal to \[49\]. So, we can write the above equation in simpler form that is
\[\Rightarrow 3{{x}^{2}}\cdot \arctan \left( 7x \right)+\dfrac{7{{x}^{3}}}{49{{x}^{2}}+1}\]
\[\therefore \] Derivative of \[{{x}^{3}}\arctan \left( 7x \right)\] is \[3{{x}^{2}}\cdot \arctan \left( 7x \right)+\dfrac{7{{x}^{3}}}{49{{x}^{2}}+1}\].

Note: Students can go wrong by not applying differentiation rule in the function \[\arctan \left( 7x \right)\] correctly that is they write \[{{\left[ \arctan \left( 7x \right) \right]}^{\prime }}=\dfrac{1}{{{\left( 7x \right)}^{2}}+1}\] and forget to multiply with the derivative of \[\left( 7x \right)\] which further leads to the wrong answer whereas correct way to write is \[{{\left[ \arctan \left( 7x \right) \right]}^{\prime }}=\dfrac{1}{{{\left( 7x \right)}^{2}}+1}\cdot \left( 7x \right)\]. So, the key point is to know both differentiation rule: \[{u}'\left( x \right)=n{{x}^{n-1}}\], \[{{\left[ \arctan \left( v\left( x \right) \right) \right]}^{\prime }}=\dfrac{1}{v{{\left( x \right)}^{2}}+1}\cdot {v}'\left( x \right)\] and the product rule: \[{{\left[ u\left( x \right).v\left( x \right) \right]}^{\prime }}=u\left( x \right).{v}'\left( x \right)+v\left( x \right).{u}'\left( x \right)\] right. And avoid multiplication, addition errors.