Question

In Exercises \(31-42,\) find \(d y / d x\). $$y=\frac{x}{\sqrt{x^{2}+1}}$$

Short answer

The derivative of the function \(y = \frac{x}{\sqrt{x^{2}+1}}\) with respect to \(x\) is \(dy/dx = \frac{1}{\sqrt{x^{2}+1}}\).

Step-by-step solution

Identify the functions

Identify the functions in the numerator and denominator. Here \(u(x) = x\) and \(v(x) = \sqrt{x^{2}+1}\). We also need their derivatives: \(u'(x) = 1\) and \(v'(x) = \frac{x}{\sqrt{x^{2}+1}}\).

Apply the Quotient Rule

Apply the quotient rule to find \(dy/dx\). According to the quotient rule, \((u/v)' = (vu' - uv')/v²\). Plugging the known values in, we get \(dy/dx = ((x * 1) - (\sqrt{x^{2}+1} * 1)) / ((\sqrt{x^{2}+1})^2)\).

Simplify the equation

Simplify the equation. Here, both the numerator and the denominator have common factor that can be cancelled. After simplification, we get \(dy/dx = \frac{1}{\sqrt{x^{2}+1}}\).