Using the limit definition, how do you differentiate #f(x)=x^3−7x+5#?
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To differentiate the function ( f(x) = x^3 - 7x + 5 ) using the limit definition of the derivative, we follow these steps:
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Start with the limit definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]
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Substitute the given function ( f(x) = x^3 - 7x + 5 ) into the limit definition: [ f'(x) = \lim_{h \to 0} \frac{(x + h)^3 - 7(x + h) + 5 - (x^3 - 7x + 5)}{h} ]
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Expand the terms in the numerator: [ f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - 7x - 7h + 5 - x^3 + 7x - 5}{h} ]
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Simplify the expression by canceling out terms: [ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 - 7h}{h} ]
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Factor out ( h ) from the numerator: [ f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2 - 7) ]
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Now, take the limit as ( h ) approaches 0: [ f'(x) = 3x^2 - 7 ]
So, the derivative of ( f(x) = x^3 - 7x + 5 ) with respect to ( x ) is ( f'(x) = 3x^2 - 7 ).
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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.
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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