Derivative of Cotangent Inverse

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

Let the function of the form

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

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

\cot y = x

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

\begin{gathered} \frac{d}{{dx}}\cot y = \frac{d}{{dx}}\left( x \right) \\ \Rightarrow - {\csc ^2}y\frac{{dy}}{{dx}} = 1 \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{1}{{{{\csc }^2}y}}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ \end{gathered}

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

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

Example: Find the derivative of

y = f\left( x \right) = {\cot ^{ - 1}}\left( {\frac{x}{a}} \right)

We have the given function as

y = {\cot ^{ - 1}}\left( {\frac{x}{a}} \right)

Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}} = \frac{d}{{dx}}{\cot ^{ - 1}}\left( {\frac{x}{a}} \right)

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

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