What is the formula for the derivative of a quotient of two functions?

The formula for the derivative of a quotient of two functions is derived from the Quotient Rule in calculus. When you have a function represented as the quotient of two differentiable functions, ( f(x) ) and ( g(x) ), the derivative is calculated using the following formula:

[

\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}

]

The numerator consists of two parts: the derivative of the numerator function ( f(x) ) multiplied by the denominator function ( g(x) ), minus the numerator function ( f(x) ) multiplied by the derivative of the denominator function ( g(x) ). This subtraction is crucial because it maintains the correct sign when applying the product of the derivatives with respect to the original functions being divided.

The denominator is simply the square of the denominator function ( g(x) ), which ensures that the entire expression represents the rate of change of the quotient accurately, considering how the function behaves near its critical points.

In summary, the correct formula captures both the rates of change of the numerator and