# Using the limit definition, how do you find the derivative of #f(x) = -7x^2 + 4x#?

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To find the derivative of ( f(x) = -7x^2 + 4x ) using the limit definition, you would first write out the definition of the derivative:

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

Then, substitute the function ( f(x) = -7x^2 + 4x ) into the formula and simplify:

[ f'(x) = \lim_{h \to 0} \frac{(-7(x + h)^2 + 4(x + h)) - (-7x^2 + 4x)}{h} ]

Expand and simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{-7(x^2 + 2xh + h^2) + 4x + 4h + 7x^2 - 4x}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{-7x^2 - 14xh - 7h^2 + 4x + 4h + 7x^2 - 4x}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{-14xh - 7h^2 + 4h}{h} ]

[ f'(x) = \lim_{h \to 0} -14x - 7h + 4 ]

[ f'(x) = -14x + 4 ]

So, the derivative of ( f(x) = -7x^2 + 4x ) with respect to ( x ) is ( f'(x) = -14x + 4 ).

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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.

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.

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.

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