# Using the limit definition, how do you find the derivative of #f(x) = -5x^2+8x+2#?

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Use the formula

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To find the derivative of ( f(x) = -5x^2 + 8x + 2 ) using the limit definition, we apply the formula:

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

First, we substitute the given function into the formula:

[ f'(x) = \lim_{h \to 0} \frac{(-5(x + h)^2 + 8(x + h) + 2) - (-5x^2 + 8x + 2)}{h} ]

Next, we expand and simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{-5(x^2 + 2xh + h^2) + 8x + 8h + 2 + 5x^2 - 8x - 2}{h} ] [ f'(x) = \lim_{h \to 0} \frac{-5x^2 - 10xh - 5h^2 + 8x + 8h + 2 + 5x^2 - 8x - 2}{h} ] [ f'(x) = \lim_{h \to 0} \frac{-10xh - 5h^2 + 8h}{h} ]

Now, we cancel out common terms and simplify further:

[ f'(x) = \lim_{h \to 0} -10x - 5h + 8 ] [ f'(x) = -10x + 8 ]

Therefore, the derivative of ( f(x) = -5x^2 + 8x + 2 ) is ( f'(x) = -10x + 8 ).

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