The Quotient Rule: Not Just Dividing Derivatives But Simple Enough
The quotient rule for derivatives allows calculation of the derivative of a function divided by another. It is essential because the derivative of a quotient of two functions isn't simply the quotient of their derivatives, necessitating a distinct formula for accurate differentiation in various applications.

Definition:
To use quotient rule, you subtract the product of the bottom function and the derivative of the top from the product of the top and the derivative of the bottom, then divide it all by the bottom function squared. Here is the mathematical formula for the quotient rule:
\( \left(\frac{f}{g}\right)’ = \frac{f’g – fg’}{g^2} \)
Example 1:
Let’s solve an example.
\( f(x) = \sin x, \ g(x) = x^2 + 1\)
\( f'(x) = \cos x, \ g'(x) = 2x \)
\(\Rightarrow \left(\frac{\sin x}{x^2 + 1}\right)’ = \frac{\cos x \cdot (x^2 + 1) – \sin x \cdot 2x}{(x^2 + 1)^2} \)
\( = \frac{\cos x \cdot x^2 + \cos x – 2x \sin x}{(x^2 + 1)^2} \)
Example 2:
\( f(x) = x^3, \ g(x) = \cos x \)
\(f'(x) = 3x^2, \ g'(x) = -\sin x \)
\(\Rightarrow \left(\frac{x^3}{\cos x}\right)’ = \frac{3x^2 \cdot \cos x – x^3 \cdot (-\sin x)}{\cos^2 x} \)
\( = \frac{3x^2 \cos x + x^3 \sin x}{\cos^2 x} \)
Hints:
- In some complex fractions, applying logarithmic differentiation simplifies the process more than the quotient rule would.
- For \( \frac{1}{x} \) and \( \frac{1}{f(x)} \) , we use the following formulas, although \( \frac{1}{x} \) could be solved using power rule too.
\( \left(\frac{1}{x}\right)’ = -\frac{1}{x^2} \)
\( \left(\frac{1}{f(x)}\right)’ = -\frac{f'(x)}{[f(x)]^2} \)
Related to This Article
More math articles
- What Kind of Math Is on the Computer Science?
- How to Compare Fractions and Decimals
- 5 Tips to Help You Learn Math Faster
- The Ultimate NYSTCE Grades 1-6 Math (222) Course (+FREE Worksheets & Tests)
- FREE 8th Grade NYSE Math Practice Test
- Overview of the Praxis Core Mathematics Test
- The Ultimate ParaPro Math Formula Cheat Sheet
- How Do You Pass the CBEST Math?
- How is the SSAT Test Scored?
- How to Graph Inverse Trigonometric Functions?
What people say about "The Quotient Rule: Not Just Dividing Derivatives But Simple Enough - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.