Derivative of Cosecant Inverse

In this tutorial we shall explore the derivative of inverse trigonometric functions and we shall prove derivative of cosecant inverse.

Let the function of the form

y = f\left( x \right) = {\csc ^{ - 1}}x

By definition of inverse trigonometric function, y = {\csc ^{ - 1}}x can be written as

\csc  y = x


Differentiating both sides with respect to the variable x, we have

\begin{gathered} \frac{d}{{dx}}\csc y = \frac{d}{{dx}}\left( x  \right) \\ \Rightarrow - \csc y\cot y\frac{{dy}}{{dx}} = 1 \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{1}{{\csc y\cot y}}\,\,\,\,{\text{  - - - }}\left( {\text{i}} \right) \\ \end{gathered}


We can write from the fundamental trigonometric rules \cot y = \sqrt  {{{\csc }^2}y - 1} . Putting this value in above relation (i) and simplifying, we have

\frac{{dy}}{{dx}}  = - \frac{1}{{\csc y\sqrt {{{\csc }^2}y  - 1} }}


But we have \csc y = x, putting this value in above relation

\begin{gathered} \frac{{dy}}{{dx}} = - \frac{1}{{x\sqrt {{x^2} - 1} }},\,\,\,\,x  \in \mathbb{R} - \left[ { - 1,1} \right] \\ \Rightarrow \frac{d}{{dx}}\left( {{{\csc }^{  - 1}}x} \right) = - \frac{1}{{x\sqrt  {{x^2} - 1} }},\,\,\,\,x \in \mathbb{R} - \left[ { - 1,1} \right] \\ \end{gathered}

Example: Find the derivative of

y = f\left( x \right)  = {\csc ^{ - 1}}{x^2}

We have the given function as

y =  {\csc ^{ - 1}}{x^2}


Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}}  = \frac{d}{{dx}}{\csc ^{ - 1}}{x^2}


Using the cosine inverse rule, \frac{d}{{dx}}\left( {{{\csc }^{ - 1}}x} \right)  = - \frac{1}{{x\sqrt {{x^2} - 1} }}, we get

\begin{gathered} \frac{{dy}}{{dx}} = - \frac{1}{{{x^2}\sqrt {{{\left( {{x^2}}  \right)}^2} - 1} }}\frac{d}{{dx}}\left( {{x^2}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{1}{{{x^2}\sqrt {{x^4} - 1} }}2x \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{{2x}}{{{x^2}\sqrt {{x^4} - 1} }} \\ \end{gathered}

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