Question

Find the derivative of sqrt(2x^2+1)

Answer 1

To find the derivative of \( y = \sqrt{2x^2 + 1} \), we will use the chain rule.

First, rewrite the expression in a form that makes differentiation easier:

\[
y = (2x^2 + 1)^{1/2}
\]

Now, we apply the chain rule. The derivative of \( y \) is:

\[
\frac{dy}{dx} = \frac{1}{2} (2x^2 + 1)^{-1/2} \cdot \frac{d}{dx}(2x^2 + 1)
\]

Next, we need to find the derivative of \( 2x^2 + 1 \):

\[
\frac{d}{dx}(2x^2 + 1) = 4x
\]

Now, substitute this back into our expression for \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = \frac{1}{2} (2x^2 + 1)^{-1/2} \cdot 4x
\]

This simplifies to:

\[
\frac{dy}{dx} = \frac{4x}{2\sqrt{2x^2 + 1}} = \frac{2x}{\sqrt{2x^2 + 1}}
\]

Thus, the derivative of \( \sqrt{2x^2 + 1} \) is:

\[
\frac{dy}{dx} = \frac{2x}{\sqrt{2x^2 + 1}}
\]