Derivative of Cosine Inverse

In this tutorial we shall be concerned with the derivative of inverse trigonometric functions and first we shall prove cosine inverse trigonometric function.

Let the function of the form

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

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

\cos y = x

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

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

Since y is restricted in the interval \left] {0,\pi } \right[ for  - 1 < x < 1, so \sin y can have only positive values, and we can write from the fundamental trigonometric rules \sin y = \sqrt {1 - {{\cos }^2}y} . Putting this value in above relation (i) and simplifying, we have

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

Example: Find the derivative of

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

We have the given function as

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

Differentiation with respect to variable x, we get

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

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

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