How is the Quotient Rule Applied in Finding Derivatives in Calculus?

The quotient rule is a fundamental concept in calculus used to find the derivative of a quotient of two functions. When differentiating a function in the form of f(x)=u(x)/v(x), the quotient rule states that the derivative is given by:

[ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2} ]

Where u'(x) and v'(x) represent the derivatives of u(x) and v(x) respectively. This rule is essential for solving complex differentiation problems involving fractions in calculus.