51 lines
3.0 KiB
Markdown
51 lines
3.0 KiB
Markdown
---
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title: Derivative
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---
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## Derivative
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**Definition** : The derivative of function f(x) with respect to x, represented by f'(x) is defined as:<br/>
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<br/><br/>
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where h is a infinitely small change in the value of input, represented by the limit function (h is approaching to zero)<br/>
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In the above formula, we notice that derivative is just the slope of a tangent of a graph of x at any input value.<br/>
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**Important property of function and it's derivative:** <br/>
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A function f(x) is differentiable at x = a, if and only if, the function is continuous at f(x=a). <br/>
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Conversely, if a derivative of a function exists at a point a, then the function must be continous at f(x=a).
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## Properties of Derivatives
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1. **Linearity**<br/>
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Suppose f(x) and g(x) are differentiable functions and a and b are real numbers. Then the function <br/>
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 <br/>
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is differentiable as <br/>
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 <br/>
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2. **Product Rule** <br/>
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For a given function h(x) = f(x) * g(x), we can apply the product rule to find the derivative of function h(x) as <br/>
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 <br/>
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Please see link in More information (Properties of Derivative) for proof of this property <br/>
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3. **Quotient Rule** <br/>
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The quotient rule gives the derivative of one function divided by another. Let h(x) = f(x) / g(x) (where g(x) cannot be zero) then the derivative of h(x) can be found using the following: <br/>
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 <br/>
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Please see link in More information (Properties of Derivative) for proof of this property <br/>
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4. **Chain Rule** <br/>
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The chain rule is used in the case of a function of a function, also known as a composite function or as a composition of functions. Input composite function representation:<br/>
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 <br/>
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Then the output derivative can be found using the following rule: <br/>
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 <br/>
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Please see link in More information (Properties of Derivative) for proof of this property <br/>
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#### More Information:
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<!-- Please add any articles you think might be helpful to read before writing the article -->
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http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeIntro.aspx
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http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx
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Properies of derivatives (proofs included) : http://www.hyper-ad.com/tutoring/math/calculus/Properties_of_Derivatives.html
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**Note** : Images taken from http://www.hyper-ad.com/ and http://tutorial.math.lamar.edu/
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