what is the derivative of x+1/x
What is the derivative of ( \frac{x+1}{x} )
Answer:
To find the derivative of the function ( f(x) = x + \frac{1}{x} ), we will use basic differentiation rules. The function ( f(x) ) is composed of two parts: the polynomial term ( x ) and the rational term ( \frac{1}{x} ).
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Differentiate ( x ):
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The derivative of ( x ) with respect to ( x ) is simply ( 1 ).
\frac{d}{dx}(x) = 1
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Differentiate ( \frac{1}{x} ):
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The term ( \frac{1}{x} ) can be written as ( x^{-1} ).
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Using the power rule ( \frac{d}{dx}(x^n) = nx^{n-1} ), we find the derivative of ( x^{-1} ):
\frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2} = -\frac{1}{x^2}
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Combine the results:
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Adding the derivatives of each term, we get:
f'(x) = \frac{d}{dx}(x + \frac{1}{x}) = \frac{d}{dx}(x) + \frac{d}{dx}(\frac{1}{x})f'(x) = 1 - \frac{1}{x^2}
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Therefore, the derivative of ( f(x) = x + \frac{1}{x} ) is:
f'(x) = 1 - \frac{1}{x^2}