Question

Calculate using quotient rule
y=1/square-root of (3x+1

Answer 1

To calculate the derivative of the function \( y = \frac{1}{\sqrt{3x + 1}} \) using the quotient rule, we first rewrite it in a form that is easier to differentiate:

\[
y = (3x + 1)^{-1/2}
\]

Now, we will differentiate using the chain rule instead of the quotient rule, as it simplifies the process.

Using the chain rule:

1. Differentiate the outer function:
\[
\frac{d}{dx}[(3x + 1)^{-1/2}] = -\frac{1}{2}(3x + 1)^{-3/2} \cdot \frac{d}{dx}(3x + 1)
\]

2. Differentiate the inner function:
\[
\frac{d}{dx}(3x + 1) = 3
\]

Putting it together:

\[
\frac{dy}{dx} = -\frac{1}{2}(3x + 1)^{-3/2} \cdot 3
\]

\[
\frac{dy}{dx} = -\frac{3}{2}(3x + 1)^{-3/2}
\]

3. If you prefer to express this with a square root in the denominator again:
\[
\frac{dy}{dx} = -\frac{3}{2\sqrt{(3x + 1)^3}}
\]

Thus, the derivative of \( y = \frac{1}{\sqrt{3x + 1}} \) is:

\[
\frac{dy}{dx} = -\frac{3}{2(3x + 1)^{3/2}}
\]