How do you find f'(x) using the definition of a derivative for #f(x)=(2x)/(2+x) #?
The definition is the following:
Expanding both numerator and denominator, we get
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To find ( f'(x) ) using the definition of a derivative for ( f(x) = \frac{{2  x}}{{2 + x}} ), follow these steps:

Use the definition of the derivative: [ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h)  f(x)}}{h} ]

Substitute ( f(x) ) into the formula: [ f'(x) = \lim_{{h \to 0}} \frac{{\frac{{2  (x + h)}}{{2 + (x + h)}}  \frac{{2  x}}{{2 + x}}}}{h} ]

Simplify the expression inside the limit: [ f'(x) = \lim_{{h \to 0}} \frac{{\frac{{2  x  h}}{{2 + x + h}}  \frac{{2  x}}{{2 + x}}}}{h} ]

Combine the fractions: [ f'(x) = \lim_{{h \to 0}} \frac{{(2  x  h)(2 + x)  (2  x)(2 + x + h)}}{{h(2 + x)(2 + x + h)}} ]

Expand and simplify the numerator: [ f'(x) = \lim_{{h \to 0}} \frac{{4  2x + 2x  x^2  2h  xh  4 + 2x + 2x^2 + 2x + xh  2x  x^2  2h  xh}}{{h(2 + x)(2 + x + h)}} ]

Cancel out terms: [ f'(x) = \lim_{{h \to 0}} \frac{{2h  2h}}{{h(2 + x)(2 + x + h)}} ]

Simplify further: [ f'(x) = \lim_{{h \to 0}} \frac{{4h}}{{h(2 + x)(2 + x + h)}} ]

Cancel out ( h ): [ f'(x) = \lim_{{h \to 0}} \frac{{4}}{{(2 + x)(2 + x + h)}} ]

Evaluate the limit as ( h ) approaches 0: [ f'(x) = \frac{{4}}{{(2 + x)(2 + x)}} ]

Simplify the expression: [ f'(x) = \frac{{4}}{{(2 + x)^2}} ]
So, ( f'(x) = \frac{{4}}{{(2 + x)^2}} ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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