Question

How do you find the derivative of $y = \sqrt {1 - {x^2}}$ ?

Answer 1

Hint : Differentiation is known as the process of dividing a whole quantity into very small ones. In this question, a function is given to us that involve the square root of x raised to some power so the given function is in terms of x, we have to differentiate $y = \sqrt {1 - {x^2}}$ with respect to x. So, the independent variable is x and the dependent variable is y. We will first differentiate the whole quantity $y = \sqrt {1 - {x^2}}$ and then differentiate the quantity in the square root as it is also a function of x $(1 - {x^2})$ . The result of multiplying these two differentiated functions will give the value of $\dfrac{{dy}}{{dx}}$ or $y'(x)$ .On solving the given question using the above information, we will get the correct answer.

Complete step-by-step answer :
We are given $y = \sqrt {1 - {x^2}}$
We know that $\dfrac{{d{x^n}}}{{dx}} = n{x^{n - 1}}$
So differentiating both sides of the above equation with respect to x, we get –
$\dfrac{{dy}}{{dx}} = \dfrac{1}{2}{(1 - {x^2})^{ - \dfrac{1}{2}}}\dfrac{{d(1 - {x^2})}}{{dx}} \\\ \dfrac{{dy}}{{dx}} = \dfrac{1}{{2\sqrt {1 - {x^2}} }}( - 2x) \\\ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{ - x}}{{\sqrt {1 - {x^2}} }} \;$
Hence, the derivative of $y = \sqrt {1 - {x^2}}$ is $\dfrac{{ - x}}{{\sqrt {1 - {x^2}} }}$
So, the correct answer is “$\dfrac{{ - x}}{{\sqrt {1 - {x^2}} }}$”.

Note : We use differentiation when we have to find the instantaneous rate of change of a quantity, it is represented as $\dfrac{{dy}}{{dx}}$ , in the expression $\dfrac{{dy}}{{dx}}$ , a very small change in quantity is represented by $dy$ and the small change in the quantity with respect to which the given quantity is changing is represented by $dx$ . We must rearrange the equation first for solving similar questions so that the variable with respect to which the function is differentiated is present on one side and the variable whose derivative we have to find is present on the other side.