How do you find f'(x) using the definition of a derivative #f(x)= 1/x#?
Use limit definition of derivative to find:
#f'(x) = -1/x^2#
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To find (f'(x)) using the definition of a derivative for (f(x) = \frac{1}{x}), you use the definition of the derivative:
[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}]
Now, simplify the expression:
[f'(x) = \lim_{h \to 0} \frac{x - (x+h)}{hx(x+h)}]
[f'(x) = \lim_{h \to 0} \frac{-h}{hx(x+h)}]
[f'(x) = \lim_{h \to 0} \frac{-1}{x(x+h)}]
[f'(x) = -\frac{1}{x^2}]
So, the derivative of (f(x) = \frac{1}{x}) with respect to (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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