# How do you use the limit definition to find the derivative of #f(x)=2/(x+4)#?

def of derivative

Substitution

Common Denominator

Distribute and write as a single numerator

Simplify

Multiply by the reciprocal

Simplify

Now we can substitute in a 0 for h

Simplify

Simplify

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To use the limit definition to find the derivative of ( f(x) = \frac{2}{x+4} ), you first apply the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute ( f(x) = \frac{2}{x+4} ) into the equation:

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

Simplify the expression:

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

[ f'(x) = \lim_{h \to 0} \frac{2x + 8 - 2x - 2h - 8}{h(x+4)(x+h+4)} ]

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

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

Now, evaluate the limit as ( h ) approaches 0:

[ f'(x) = \frac{-2}{(x+4)^2} ]

So, the derivative of ( f(x) = \frac{2}{x+4} ) is ( f'(x) = \frac{-2}{(x+4)^2} ).

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