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How do you find the derivative using limits of #f(x)=3x+2#?

Answer 1

Luke Alvarez

Use the definition of a derivative

Consider below:

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Answer 2

Ezra Adams

To find the derivative of ( f(x) = 3x + 2 ) using limits, you apply the definition of the derivative:

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

Substitute ( f(x) = 3x + 2 ) into the equation:

[ f'(x) = \lim_{h \to 0} \frac{3(x + h) + 2 - (3x + 2)}{h} ]

Simplify:

[ f'(x) = \lim_{h \to 0} \frac{3x + 3h + 2 - 3x - 2}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{3h}{h} ]

[ f'(x) = \lim_{h \to 0} 3 ]

[ f'(x) = 3 ]

So, the derivative of ( f(x) = 3x + 2 ) is ( f'(x) = 3 ).

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Answer from HIX Tutor

When evaluating a one-sided 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.