Using the quotient rule to find the derivative

 
 
Quotient rule blog post.jpeg
 
 
 

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.

Krista King Math.jpg

Hi! I'm krista.

I create online courses to help you rock your math class. Read more.

 

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


 
Krista King Math Signup.png
 
Calculus 1 course.png

Take the course

Want to learn more about Calculus 1? I have a step-by-step course for that. :)

 
 

 
 

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???.

Quotient rule for Calculus 1.jpg

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.

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}???

 
Krista King.png
 

Get access to the complete Calculus 1 course