How do you use the definition of a derivative to find the derivative of #f(x) =1/x#?
When you are asked to differentiate using the definition of the derivative, you need to use the formula
Using the power rule to confirm:
Hopefully this helps!
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To find the derivative of (f(x) = \frac{1}{x}) using the definition of a derivative, we start with the definition:
[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}]
Substitute (f(x) = \frac{1}{x}) into the definition:
[f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h}]
Combine the fractions in the numerator:
[f'(x) = \lim_{h \to 0} \frac{\frac{x - (x + h)}{x(x + h)}}{h}]
Simplify the numerator:
[f'(x) = \lim_{h \to 0} \frac{-h}{x(x + h)h}]
Cancel out the (h) in the numerator and denominator:
[f'(x) = \lim_{h \to 0} \frac{-1}{x(x + h)}]
Take the limit as (h) approaches 0:
[f'(x) = \frac{-1}{x^2}]
So, the derivative of (f(x) = \frac{1}{x}) is (f'(x) = \frac{-1}{x^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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