# How do you find the derivative of sqrt(4+x^2)?

Jul 12, 2016

Reqd. Deri. $= \frac{x}{\sqrt{4 + {x}^{2}}} .$

#### Explanation:

Let $y = \sqrt{4 + {x}^{2}} , = \sqrt{t} \ldots \ldots \ldots \left(1\right)$, where $t = 4 + {x}^{2.} \ldots \left(2\right)$

Thus, $y$ is a fun. of $t$, $t$ that of $x$.

In such cases, the Chain Rule is to be applied to get the reqd. deri.

Reqd. Deri. $= \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}$
$= \frac{d}{\mathrm{dt}} \left(\sqrt{t}\right) \cdot \frac{d}{\mathrm{dx}} \left(4 + {x}^{2}\right)$............[$\left(1\right) \mathmr{and} \left(2\right)$]
$= \frac{1}{2 \sqrt{t}} \left(2 x\right) = \frac{x}{\sqrt{t}} = \frac{x}{\sqrt{4 + {x}^{2}}} .$