Chapter 4.1, Problem 20E

To determine

To calculate:The differentiation of the given function.

Answer to Problem 20E

The differentiation is $\frac{1}{{(1+x^2)}^{\frac{3}{2}}}$ .

Explanation of Solution

Given information: Given function is $y=\frac{x}{\sqrt{1+x^2}}$ .

Formula used: $\frac{d(x^n)}{dx}=n{x}^{n-1}$ with quotient, chain and addition rule.

Calculation:

  $y=\frac{x}{\sqrt{1+x^2}}$

Differentiate with respect to x :

  $\begin{array}{l}\frac{dy}{dx}=\frac{\frac{d(x)}{dx}\times \sqrt{1+x^2}-x\times \frac{d(\sqrt{1+x^2)}}{d(1+x^2)}\times \frac{d(1+x^2)}{dx}}{{(\sqrt{1+x^2})}^2}\\ \frac{dy}{dx}=\frac{\sqrt{1+x^2}-x\times \frac{1}{2\sqrt{1+x^2}}\times 2x}{(1+x^2)}\\ \frac{dy}{dx}=\frac{\sqrt{1+x^2}-\frac{x^2}{\sqrt{1+x^2}}}{(1+x^2)}\\ \frac{dy}{dx}=\frac{1+x^2-x^2}{\sqrt{1+x^2}(1+x^2)}\\ \frac{dy}{dx}=\frac{1}{{(1+x^2)}^{\frac{3}{2}}}\end{array}$