Difference between revisions of "Derivatives of Products and Quotients"
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* <math> f(x) = \sqrt{x} </math>, <math> g(x) = \sin{x} </math>. <math> f'(x) = \frac{1}{2\sqrt{x}} </math> and <math> g'(x) = \cos{x} </math>, so <math> \frac{d}{dx}(\sqrt{x})(\sin{x}) = (\frac{1}{2\sqrt{x}})(\cos{x}) + \sqrt{x}\sin{x}</math>. | * <math> f(x) = \sqrt{x} </math>, <math> g(x) = \sin{x} </math>. <math> f'(x) = \frac{1}{2\sqrt{x}} </math> and <math> g'(x) = \cos{x} </math>, so <math> \frac{d}{dx}(\sqrt{x})(\sin{x}) = (\frac{1}{2\sqrt{x}})(\cos{x}) + \sqrt{x}\sin{x}</math>. | ||
+ | ===Quotient Rule=== | ||
+ | Let <math> f(x) </math> and <math> g(x) </math> be differentiable functions such that <math> g(x) \neq 0 </math>. Then, the derivative of the quotient, <math> \frac{f(x)}{g(x)} </math>, is | ||
− | = | + | <math> \frac{d}{dx}\(\frac{f(x)}{g(x)}\) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} </math>. |
==Resources== | ==Resources== | ||
* [http://www2.gcc.edu/dept/math/faculty/BancroftED/buscalc/chapter2/section2-4.php Product and Quotient Rules], Grover City College | * [http://www2.gcc.edu/dept/math/faculty/BancroftED/buscalc/chapter2/section2-4.php Product and Quotient Rules], Grover City College | ||
* [https://tutorial.math.lamar.edu/classes/calci/productquotientrule.aspx Product and Quotient Rule], Paul's Online Notes (Lamar University) | * [https://tutorial.math.lamar.edu/classes/calci/productquotientrule.aspx Product and Quotient Rule], Paul's Online Notes (Lamar University) |
Revision as of 10:31, 21 September 2021
Product Rule
Let and be differentiable functions. Then, the derivative of their product, , is
.
Examples:
- , . and , so .
- , . and , so .
Quotient Rule
Let and be differentiable functions such that . Then, the derivative of the quotient, , is
Failed to parse (syntax error): {\displaystyle \frac{d}{dx}\(\frac{f(x)}{g(x)}\) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} } .
Resources
- Product and Quotient Rules, Grover City College
- Product and Quotient Rule, Paul's Online Notes (Lamar University)