Quotient rule

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let f ( x ) = g ( x ) / h ( x ) , {displaystyle f(x)=g(x)/h(x),} where both g {displaystyle g} and h {displaystyle h} are differentiable and h ( x ) ≠ 0. {displaystyle h(x) eq 0.} The quotient rule states that the derivative of f ( x ) {displaystyle f(x)} is In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let f ( x ) = g ( x ) / h ( x ) , {displaystyle f(x)=g(x)/h(x),} where both g {displaystyle g} and h {displaystyle h} are differentiable and h ( x ) ≠ 0. {displaystyle h(x) eq 0.} The quotient rule states that the derivative of f ( x ) {displaystyle f(x)} is Let f ( x ) = g ( x ) / h ( x ) . {displaystyle f(x)=g(x)/h(x).} Applying the definition of the derivative and properties of limits gives the following proof. Let f ( x ) = g ( x ) h ( x ) , {displaystyle f(x)={frac {g(x)}{h(x)}},} so g ( x ) = f ( x ) h ( x ) . {displaystyle g(x)=f(x)h(x).} The product rule then gives g ′ ( x ) = f ′ ( x ) h ( x ) + f ( x ) h ′ ( x ) . {displaystyle g'(x)=f'(x)h(x)+f(x)h'(x).} Solving for f ′ ( x ) {displaystyle f'(x)} and substituting back for f ( x ) {displaystyle f(x)} gives: Let f ( x ) = g ( x ) h ( x ) = g ( x ) h ( x ) − 1 . {displaystyle f(x)={frac {g(x)}{h(x)}}=g(x)h(x)^{-1}.} Then the product rule gives To evaluate the derivative in the second term, apply the power rule along with the chain rule: