How do you use the limit definition of the derivative to find the derivative of #f(x)=x^3+2x#?
By definition:
so:
Expand the cube of the binomial and simplify:
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To find the derivative of ( f(x) = x^3 + 2x ) using the limit definition of the derivative, follow these steps:
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Write down the limit definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]
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Substitute ( f(x) = x^3 + 2x ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(x + h)^3 + 2(x + h) - (x^3 + 2x)}{h} ]
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Expand ( (x + h)^3 ) and simplify: [ (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 ] [ f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 + 2x + 2h - x^3 - 2x}{h} ]
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Cancel out like terms: [ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 + 2h}{h} ]
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Factor out an ( h ) from the numerator: [ f'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2) + 2h}{h} ]
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Cancel out an ( h ) from the numerator and denominator: [ f'(x) = \lim_{h \to 0} 3x^2 + 3xh + h^2 + 2 ]
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Evaluate the limit as ( h ) approaches 0: [ f'(x) = 3x^2 + 2 ]
So, the derivative of ( f(x) = x^3 + 2x ) is ( f'(x) = 3x^2 + 2 ).
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To use the limit definition of the derivative to find the derivative of ( f(x) = x^3 + 2x ), follow these steps:
- Write down the limit definition of the derivative:
[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]
- Substitute the function ( f(x) = x^3 + 2x ) into the definition:
[ f'(x) = \lim_{h \to 0} \frac{(x+h)^3 + 2(x+h) - (x^3 + 2x)}{h} ]
- Expand ( (x+h)^3 ) using the binomial theorem:
[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 ]
- Substitute the expanded expression into the limit expression:
[ f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 + 2(x+h) - (x^3 + 2x)}{h} ]
- Simplify the expression by canceling out common terms:
[ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 + 2h}{h} ]
- Factor out ( h ) from the numerator:
[ f'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2) + 2h}{h} ]
- Cancel out ( h ) from the numerator and denominator:
[ f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2 + 2) ]
- Now, substitute ( h = 0 ) into the expression:
[ f'(x) = 3x^2 + 2 ]
So, the derivative of ( f(x) = x^3 + 2x ) is ( f'(x) = 3x^2 + 2 ).
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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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