How do you use the definition of the derivative to differentiate the function #f(x)= 2x^25#?
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To differentiate the function ( f(x) = 2x^2  5 ) using the definition of the derivative, follow these steps:

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

Substitute the function ( f(x) = 2x^2  5 ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(2(x+h)^2  5)  (2x^2  5)}{h} ]

Expand and simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{2(x^2 + 2hx + h^2)  5  2x^2 + 5}{h} ] [ f'(x) = \lim_{h \to 0} \frac{2x^2 + 4hx + 2h^2  2x^2}{h} ] [ f'(x) = \lim_{h \to 0} \frac{4hx + 2h^2}{h} ]

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

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

Evaluate the limit as ( h ) approaches 0: [ f'(x) = 4x ]
Therefore, the derivative of the function ( f(x) = 2x^2  5 ) with respect to ( x ) is ( f'(x) = 4x ).
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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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