Derivative of sqrt(1 - x^2)

The calculator will find the derivative of

\sqrt{1 - x^{2}}

, with steps shown.

Your Input

Find $\frac{d}{dx} \left(\sqrt{1 - x^{2}}\right)$ .

Solution

The function $\sqrt{1 - x^{2}}$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = \sqrt{u}$ and $g{\left(x \right)} = 1 - x^{2}$ .

Apply the chain rule $\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$ :

{\color{red}\left(\frac{d}{dx} \left(\sqrt{1 - x^{2}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sqrt{u}\right) \frac{d}{dx} \left(1 - x^{2}\right)\right)}

Apply the power rule $\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$ with $n = \frac{1}{2}$ :

{\color{red}\left(\frac{d}{du} \left(\sqrt{u}\right)\right)} \frac{d}{dx} \left(1 - x^{2}\right) = {\color{red}\left(\frac{1}{2 \sqrt{u}}\right)} \frac{d}{dx} \left(1 - x^{2}\right)

Return to the old variable:

\frac{\frac{d}{dx} \left(1 - x^{2}\right)}{2 \sqrt{{\color{red}\left(u\right)}}} = \frac{\frac{d}{dx} \left(1 - x^{2}\right)}{2 \sqrt{{\color{red}\left(1 - x^{2}\right)}}}

The derivative of a sum/difference is the sum/difference of derivatives:

\frac{{\color{red}\left(\frac{d}{dx} \left(1 - x^{2}\right)\right)}}{2 \sqrt{1 - x^{2}}} = \frac{{\color{red}\left(\frac{d}{dx} \left(1\right) - \frac{d}{dx} \left(x^{2}\right)\right)}}{2 \sqrt{1 - x^{2}}}

The derivative of a constant is $0$ :

\frac{{\color{red}\left(\frac{d}{dx} \left(1\right)\right)} - \frac{d}{dx} \left(x^{2}\right)}{2 \sqrt{1 - x^{2}}} = \frac{{\color{red}\left(0\right)} - \frac{d}{dx} \left(x^{2}\right)}{2 \sqrt{1 - x^{2}}}

Apply the power rule $\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$ with $n = 2$ :

- \frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)}}{2 \sqrt{1 - x^{2}}} = - \frac{{\color{red}\left(2 x\right)}}{2 \sqrt{1 - x^{2}}}

Thus, $\frac{d}{dx} \left(\sqrt{1 - x^{2}}\right) = - \frac{x}{\sqrt{1 - x^{2}}}$ .

Answer

$\frac{d}{dx} \left(\sqrt{1 - x^{2}}\right) = - \frac{x}{\sqrt{1 - x^{2}}}$ A

Derivative of 1−x2\sqrt{1 - x^{2}}1−x2​

Your Input

Solution

Answer

Derivative of $\sqrt{1 - x^{2}}$