Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at $\left( 2 , \frac { 1 } { 2 } \right)$ .
A) $y = - \frac { x } { 4 } + 1$
B) $y = - \frac { x } { 2 } + 1$
C) y = -x + 1
D) $y = \frac { x } { 2 } + 1$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line, first, calculate the derivative of $f(x) = \frac{1}{x}$, which is $f'(x) = -\frac{1}{x^2}$. Evaluating this derivative at $x = 2$ gives $f'(2) = -\frac{1}{4}$, which is the slope of the tangent line. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with $m = -\frac{1}{4}$ and the point $(2, \frac{1}{2})$, we get $y - \frac{1}{2} = -\frac{1}{4}(x - 2)$. Simplifying this equation gives $y = -\frac{x}{4} + 1$, which matches choice A.

Find the Equation of the Tangent Line to the Graph f(x)=1xf ( x ) = \frac { 1 } { x }f(x)=x1​

Find the Equation of the Tangent Line to the Graph $f ( x ) = \frac { 1 } { x }$