Questions: If (f(x)=frac7-x^23+x^2), find: [ f^prime(x)= ]

$If (f(x)=frac7-x^23+x^2), find: [ f^prime(x)= ]$

Transcript text: If $f(x)=\frac{7-x^{2}}{3+x^{2}}$, find: \[ f^{\prime}(x)= \] Question Help: Video Submit Question

Solution

Solution Steps

Step 1: Identify the function and its components

To find the derivative of the function $f(x) = \frac{7 - x^2}{3 + x^2}$, we identify $u(x) = 7 - x^2$ and $v(x) = 3 + x^2$.

Step 2: Apply the quotient rule

The quotient rule for differentiation is given by $g'(x) = \frac{{u'(x)v(x) - u(x)v'(x)}}{{[v(x)]^2}}$. Here, $u'(x) = -2x$ and $v'(x) = 2x$. Substituting these into the quotient rule gives: $f'(x) = \frac{(-2x)(3 + x^2) - (7 - x^2)(2x)}{(3 + x^2)^2}$.

Step 3: Simplify the expression

Simplifying the expression, we get: $f'(x) = \frac{-2x(3 + 7)}{(3 + x^2)^2}$.