Question

How to find the derivative of \[{{\sin }^{-1}}\left( x \right)\]?

Answer 1

Hint: In the above question where we need to find the derivative of \[{{\sin }^{-1}}\left( x \right)\] we will first assume a variable and then differentiate with respect to that assumed variable and then convert that differentiated value in terms of assumed value which will then convert to give us the final answer.

Complete step by step answer:
In the above type of question we need to first assume a variable. So, Let \[y={{\sin }^{-1}}\left( x \right)\]
Now we will take sin on both sides and as we know that \[\sin \left( {{\sin }^{-1}}\left( x \right) \right)=x\] we will use this property of sin while differentiating. So on taking sin on both the sides we will get it as:
\[\begin{align}
  & \Rightarrow \sin \left( y \right)=\sin \left( {{\sin }^{-1}}\left( x \right) \right) \\
 & \Rightarrow \sin \left( y \right)=x \\
\end{align}\]
Now that the inverse of sin is removed and with basic differentiating properties we can differentiate the above function. So on differentiating both the sides of the inequality with respect to x we will get it as:
\[\begin{align}
  & \Rightarrow \sin \left( y \right)=\sin \left( {{\sin }^{-1}}\left( x \right) \right) \\
 & \Rightarrow \cos \left( y \right)\left( \dfrac{dy}{dx} \right)=1 \\
\end{align}\]
This can also be written as :
\[\Rightarrow \left( \dfrac{dy}{dx} \right)=\dfrac{1}{\cos \left( y \right)}\]
Now that we know the value of derivative of the given function but the function is in the form of assumed function so to remove those we will use another property of sin and cosine which is as follows:
\[{{\sin }^{2}}\left( x \right)+{{\cos }^{2}}\left( x \right)=1\]
Now with this we can clearly say that \[\cos \left( x \right)=\sqrt{1-{{\sin }^{2}}\left( x \right)}\] and now substituting this in the given equation we can say that:
\[\Rightarrow \left( \dfrac{dy}{dx} \right)=\dfrac{1}{\sqrt{1-{{\sin }^{2}}\left( y \right)}}\]
Now we will substitute \[y={{\sin }^{-1}}\left( x \right)\] in the above equation and as we know that \[\sin \left( {{\sin }^{-1}}\left( x \right) \right)=x\] we can use this property to get to the final answer and by substituting the value of y and then using the property we will get the answer as:
\[\begin{align}
  & \Rightarrow \left( \dfrac{dy}{dx} \right)=\dfrac{1}{\sqrt{1-{{\sin }^{2}}\left( {{\sin }^{-1}}\left( x \right) \right)}} \\
 & \Rightarrow \left( \dfrac{dy}{dx} \right)=\dfrac{1}{\sqrt{1-{{x}^{2}}}} \\
\end{align}\]

So we can say that the derivative of \[{{\sin }^{-1}}\left( x \right)\] is \[\dfrac{1}{\sqrt{1-{{x}^{2}}}}\].

Note: In the above type of questions when you don’t know what can be done to solve this problem always try to make assumptions which will make the equation of question easier and by using those properties we can easily get to the final answer. There is also a small mistake that happens that we forget to change the assumed value to the value given in the question so try to write the answer with no assumed variable will be present in the final answer.