Question

What is the derivative of $y=\arccos \left( x \right)$ ?

Answer 1

To solve the question given above we will use the chain rule which can be written as ${F}'\left( x \right)={f}'\left( g\left( x \right) \right){g}'\left( x \right)$ and then simplify and evaluate the expression by applying the derivative rules as required to get the required and final solution.

Complete step by step solution:
The given function whose derivative we have to find is $y=\arccos \left( x \right)$ …$\left( i \right)$
Let us now multiply both the sides of the above equation with $\cos$ to get,
$\Rightarrow \cos y=\cos \left( arc\cos \left( x \right) \right)$
We also know that $\arccos \left( x \right)$ can also be written using the inverse notation ${{\sin }^{-1}}$ , using this we have the property that $\cos \left( \arccos \left( x \right) \right)=x$ . The same concept can be applied to $\arcsin \left( x \right)$ as well.
From the above equation, after simplifying by using the above mentioned property we get,
$\Rightarrow \cos y=x$
We know that on differentiating a constant we get zero and on differentiating a variable with the coefficient as one, we get its derivative as one.
Now, we will use implicit differentiation on the above equation, while also applying the chain rule on $\cos y$ because we have to differentiate with respect to $x$, to get,
$\Rightarrow -\sin y\dfrac{dy}{dx}=1$
Now, we divide both the sides of the above equation by $-\sin y$, to get,
$\Rightarrow \dfrac{dy}{dx}=-\dfrac{1}{\sin y}$
Now, we upon substituting equation $\left( i \right)$ in the above equation, we get the following expression,
$\Rightarrow \dfrac{dy}{dx}=-\dfrac{1}{\sin \left( \arccos \left( x \right) \right)}$
We can simplify the above expression by using the identity $\sin \left( \arccos \left( x \right) \right)=\cos \left( \arcsin \left( x \right) \right)=\sqrt{1-{{x}^{2}}}$ . Therefore, on applying the given identity to the denominator of the above equation, we get,
$\Rightarrow \dfrac{dy}{dx}=-\dfrac{1}{\sqrt{1-{{x}^{2}}}}$
Therefore, on differentiating the given function $y=\arccos \left( x \right)$ we get our final answer as $\dfrac{dy}{dx}=-\dfrac{1}{\sqrt{1-{{x}^{2}}}}$

Note: While solving these types of questions, students should be aware of basic differentiation properties and laws. Some differentiation laws and rules such as the chain rule and quotient rule can be very useful while solving different problems.