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Differentiate \[\left[ {{{\tan }^{ - 1}}\left( {\dfrac{x}{{\sqrt {{a^2} - {x^2}} }}} \right)} \right]\], with respect to x.

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Last updated date: 13th Jul 2024
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Answer
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Hint- Here, we will proceed by simplifying the given function (which needs to be differentiated) with the help of a proper substitution.

Let the function be \[y = \left[ {{{\tan }^{ - 1}}\left( {\dfrac{x}{{\sqrt {{a^2} - {x^2}} }}} \right)} \right]\]
Put $x = a\sin \theta $ in the above function, we get
\[ \Rightarrow y = \left[ {{{\tan }^{ - 1}}\left( {\dfrac{{a\sin \theta }}{{\sqrt {{a^2} - {{\left( {a\sin \theta } \right)}^2}} }}} \right)} \right] = \left[ {{{\tan }^{ - 1}}\left( {\dfrac{{a\sin \theta }}{{\sqrt {{a^2}\left[ {1 - {{\left( {\sin \theta } \right)}^2}} \right]} }}} \right)} \right] = \left[ {{{\tan }^{ - 1}}\left( {\dfrac{{a\sin \theta }}{{a\sqrt {1 - {{\left( {\sin \theta } \right)}^2}} }}} \right)} \right]\]
As, we know that ${\left( {\sin \theta } \right)^2} + {\left( {\cos \theta } \right)^2} = 1 \Rightarrow 1 - {\left( {\sin \theta } \right)^2} = {\left( {\cos \theta } \right)^2}$
Using this above identity to simplify the function whose differentiation needs to be carried out
\[
   \Rightarrow y = \left[ {{{\tan }^{ - 1}}\left( {\dfrac{{a\sin \theta }}{{a\sqrt {1 - {{\left( {\sin \theta } \right)}^2}} }}} \right)} \right] = \left[ {{{\tan }^{ - 1}}\left( {\dfrac{{\sin \theta }}{{\sqrt {{{\left( {\cos \theta } \right)}^2}} }}} \right)} \right] = \left[ {{{\tan }^{ - 1}}\left( {\dfrac{{\sin \theta }}{{\cos \theta }}} \right)} \right] = \left[ {{{\tan }^{ - 1}}\left( {\tan \theta } \right)} \right] = \theta \\
   \Rightarrow y = \theta {\text{ }} \to {\text{(1)}} \\
 \]
Also, we know that $x = a\sin \theta \Rightarrow \sin \theta = \dfrac{x}{a} \Rightarrow \theta = {\sin ^{ - 1}}\left( {\dfrac{x}{a}} \right){\text{ }} \to {\text{(2)}}$
Substituting the value obtained from equation (2) in equation (1), we get
\[ \Rightarrow y = {\sin ^{ - 1}}\left( {\dfrac{x}{a}} \right){\text{ }} \to {\text{(3)}}\]
Also we know that $\dfrac{{d\left[ {{{\sin }^{ - 1}}\left( {f\left( x \right)} \right)} \right]}}{{dx}} = \left[ {\dfrac{1}{{\sqrt {1 - {{\left( {f\left( x \right)} \right)}^2}} }}} \right] \times \left[ {\dfrac{{d\left( {f\left( x \right)} \right)}}{{dx}}} \right]$
Differentiating both sides of equation (3) with respect to $x$, we get
\[
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{d\left[ {{{\sin }^{ - 1}}\left( {\dfrac{x}{a}} \right)} \right]}}{{dx}}{\text{ = }}\left[ {\dfrac{1}{{\sqrt {1 - {{\left( {\dfrac{x}{a}} \right)}^2}} }}} \right] \times \left[ {\dfrac{{d\left( {\dfrac{x}{a}} \right)}}{{dx}}} \right] = \left[ {\dfrac{1}{{\sqrt {\left( {\dfrac{{{a^2} - {x^2}}}{{{a^2}}}} \right)} }}} \right] \times \left[ {\dfrac{1}{a}} \right] = \left[ {\dfrac{a}{{\sqrt {{a^2} - {x^2}} }}} \right] \times \left[ {\dfrac{1}{a}} \right] \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{\sqrt {{a^2} - {x^2}} }} \\
 \]
Therefore, the differentiation of the given function \[\left[ {{{\tan }^{ - 1}}\left( {\dfrac{x}{{\sqrt {{a^2} - {x^2}} }}} \right)} \right]\], with respect to $x$ is \[\dfrac{1}{{\sqrt {{a^2} - {x^2}} }}\].

Note- In this problem we have especially substituted $x = a\sin \theta $ because due to this substitution, the function inside the tangent inverse will be converted into tangent so that these two will cancel out with each other and we will be left with the angle which is in a much simplified form.