The derivative of root 1-x^2 is equal to -x/√(1-x^{2}). Here we differentiate square root of 1-x^{2} by chain rule and implicit differentiation method.

Note that

$\dfrac{d}{dx} \Big( \sqrt{1-x^2}\Big)=-\dfrac{x}{\sqrt{1-x^2}}$.

Let us now differentiate root 1-x^{2}.

## Derivative of root 1-x^{2} by Chain Rule

**Question:** Find the derivative of square root 1-x^{2}.

**Answer:**

Let us put z = $1-x^2$.

So that $\dfrac{dz}{dx}=-2x$.

By the chain rule, the derivative of root 1-x^{2} is equal to

$\dfrac{d}{dx} \Big( \sqrt{1-x^2}\Big)= \dfrac{d}{dx} ( \sqrt{z})$

= $\dfrac{d}{dz} ( \sqrt{z}) \times \dfrac{dz}{dx}$

= $\dfrac{d}{dz}$ (z^{1/2}) × (-2x) as dz/dx= -2x.

= $-2x \times \dfrac{1}{2} z^{1/2 -1}$

= $-\dfrac{x}{z^{1/2 }}$

= $-\dfrac{x}{\sqrt{1-x^2}}$ as z=1-x^{2}.

So the derivative of square root 1-x^{2} by the chain rule is equal to -x/√(1-x^{2}).

## Derivative of root 1-x^{2} by Implicit Differentiation

**Answer:** The differentiation of square root of 1-x^{2} is equal to -x/√(1-x^{2}).

**Explanation:**

We will use the implicit differentiation method to find the derivative of square root of 1-x square. So let us put

y = $\sqrt{1-x^2}$.

Squaring both sides, we get that

y^{2} = 1-x^{2}.

Differentiating both sides w.r.t x, we get that

$2y \dfrac{dy}{dx}=-2x$

⇒ $\dfrac{dy}{dx}=-\dfrac{x}{y}$

⇒ $\dfrac{dy}{dx}=-\dfrac{x}{\sqrt{1-x^2}}$, putting the value of y.

So the derivative of square root of 1-x^2 is equal to -x/√(1-x^{2}), and this is obtained by the implicit differentiation method.

**Also Read:** Derivative of root x

## FAQs

### Q1: What is the derivative of root 1-x^{2}?

**Answer:** The derivative of root 1-x^{2} is equal to -x/√(1-x^{2}).