Using the quotient rule to find the derivative
The quotient rule differentiates a quotient, just like the product rule differentiates a product
Just as you must always use the product rule when two variable expressions are multiplied, you must use the quotient rule whenever two variable expressions are divided.
Given a function
???h(x)=\frac{f(x)}{g(x)}???
then its derivative is
???h'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{\left[g(x)\right]^2}???
Applying the quotient rule formula to find the derivative
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Using quotient rule with power and log functions
Example
Use quotient rule to find the derivative.
???h(x)=\frac{x^2}{\ln{x}}???
Based on the quotient rule formula, we know that ???f(x)??? is the numerator and therefore ???f(x)=x^2??? and that ???g(x)??? is the denominator and therefore that ???g(x)=\ln{x}???. ???f'(x)=2x???, and ???g'(x)=1/x???.
Plugging all of these components into the quotient rule gives
???h'(x)=\frac{\left(\ln{x}\right)(2x)-\left(x^2\right)\left(\frac{1}{x}\right)}{\left(\ln{x}\right)^2}???
???h'(x)=\frac{2x\ln{x}-x}{\left(\ln{x}\right)^2}???