How do you use the definition of a derivative to show that if #f(x)=1/x# then #f'(x)=-1/x^2#?
This is a proof
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To use the definition of a derivative to show that if ( f(x) = \frac{1}{x} ), then ( f'(x) = -\frac{1}{x^2} ), we start with the definition of the derivative:
[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]
Substituting ( f(x) = \frac{1}{x} ) into this definition, we get:
[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} ]
Now, we need to simplify this expression. To do so, we find a common denominator:
[ f'(x) = \lim_{h \to 0} \frac{x - (x+h)}{x(x+h)h} ]
[ f'(x) = \lim_{h \to 0} \frac{x - x - h}{x(x+h)h} ]
[ f'(x) = \lim_{h \to 0} \frac{-h}{x(x+h)h} ]
Now, we cancel out ( h ) from the numerator and denominator:
[ f'(x) = \lim_{h \to 0} \frac{-1}{x(x+h)} ]
As ( h ) approaches 0, ( x+h ) approaches ( x ), so we can substitute ( x ) for ( x+h ):
[ f'(x) = \frac{-1}{x \cdot x} ]
[ f'(x) = -\frac{1}{x^2} ]
Thus, we have shown that if ( f(x) = \frac{1}{x} ), then ( f'(x) = -\frac{1}{x^2} ) using the definition of a derivative.
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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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