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.
[include_netrun_products_block from-products="product/6-south-carolina-sc-ready-grade-3-math-practice-tests/" product-list-class="bundle-products float-left" product-item-class="float-left" product-item-image-container-class="p-0 float-left" product-item-image-container-size="col-2" product-item-image-container-custom-style="" product-item-container-size="" product-item-add-to-cart-class="btn-accent btn-purchase-ajax" product-item-button-custom-url="{url}/?ajax-add-to-cart={id}" product-item-button-custom-url-if-not-salable="{productUrl} product-item-container-class="" product-item-element-order="image,title,purchase,price" product-item-title-size="" product-item-title-wrapper-size="col-10" product-item-title-tag="h3" product-item-title-class="mt-0" product-item-title-wrapper-class="float-left pr-0" product-item-price-size="" product-item-purchase-size="" product-item-purchase-wrapper-size="" product-item-price-wrapper-class="pr-0 float-left" product-item-price-wrapper-size="col-10" product-item-read-more-text="" product-item-add-to-cart-text="" product-item-add-to-cart-custom-attribute="title='Purchase this book with single click'" product-item-thumbnail-size="290-380" show-details="false" show-excerpt="false" paginate="false" lazy-load="true"] [include_netrun_products_block from-products="product/6-south-carolina-sc-ready-grade-3-math-practice-tests/" product-list-class="bundle-products float-left" product-item-class="float-left" product-item-image-container-class="p-0 float-left" product-item-image-container-size="col-2" product-item-image-container-custom-style="" product-item-container-size="" product-item-add-to-cart-class="btn-accent btn-purchase-ajax" product-item-button-custom-url="{url}/?ajax-add-to-cart={id}" product-item-button-custom-url-if-not-salable="{productUrl} product-item-container-class="" product-item-element-order="image,title,purchase,price" product-item-title-size="" product-item-title-wrapper-size="col-10" product-item-title-tag="h3" product-item-title-class="mt-0" product-item-title-wrapper-class="float-left pr-0" product-item-price-size="" product-item-purchase-size="" product-item-purchase-wrapper-size="" product-item-price-wrapper-class="pr-0 float-left" product-item-price-wrapper-size="col-10" product-item-read-more-text="" product-item-add-to-cart-text="" product-item-add-to-cart-custom-attribute="title='Purchase this book with single click'" product-item-thumbnail-size="290-380" show-details="false" show-excerpt="false" paginate="false" lazy-load="true"]
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
- ACT Test Calculator Policy
- 7th Grade GMAS Math Worksheets: FREE & Printable
- Intelligent Math Puzzle – Challenge 89
- 5th Grade PSSA Math Practice Test Questions
- How to Use Memory Tricks to Memorize Math Formulas?
- SAT Math Vs. High School Math
- Geometry Puzzle – Challenge 70
- 5th Grade IAR Math Worksheets: FREE & Printable
- How to Find Midpoint? (+FREE Worksheet!)
- How to Find 3D Shapes Nets?
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.