,
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,
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Find the equation of the tangent line of the function
at the point .
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Let .
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A)If , find .
B) Find .
You need to know that $\displaystyle \frac{d}{dx}ln(x) = \frac{1}{x} = x^{-1}$.
So
$\displaystyle f(x) = x^{-1} + x^{-2} + ln(x)$
$\displaystyle f^{\prime}(x) = (-1)x^{-2} + (-2)x^{-3} + x^{-1}$
So
$\displaystyle f^{\prime}(1) = -(1)^{-2} - 2(1)^{-3} + (1)^{-1} = -2$
So the slope of the tangent line to the function f(x) at x = 1 is -2.
What point is the tangent line passing through? Well, it passes through the point on the function at x = 1:
$\displaystyle f(1) = (1)^{-1} + (1)^{-2} + ln(1) = 2$
So we want a line with a slope of -2 that passes through the point (1, 2).
I leave the rest to you.
-Dan