Differentiate \(\frac{x}{x + 1}\) with respect to x.
Answer Details
To differentiate \(\frac{x}{x + 1}\) with respect to x, we can use the quotient rule of differentiation, which states that for functions u(x) and v(x), the derivative of \(\frac{u(x)}{v(x)}\) is given by:
\[\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x) v(x) - u(x) v'(x)}{(v(x))^2}\]
Applying this rule to the given function, we have:
\[u(x) = x\]
\[v(x) = x + 1\]
So, we need to find u'(x) and v'(x):
\[u'(x) = 1\]
\[v'(x) = 1\]
Substituting these values into the quotient rule formula, we get:
\[\frac{d}{dx} \left( \frac{x}{x + 1} \right) = \frac{1(x + 1) - x(1)}{(x + 1)^2}\]
Simplifying the numerator and denominator, we get:
\[\frac{d}{dx} \left( \frac{x}{x + 1} \right) = \frac{1}{(x + 1)^2}\]
Therefore, the correct answer is \(\frac{1}{(x + 1)^2}\).