Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at the point $\left( 6 , \frac { 1 } { 6 } \right)$ .
A) $y = - \frac { 1 } { 36 } x + \frac { 1 } { 3 }$
B) $y = - \frac { 1 } { 6 } x + \frac { 1 } { 3 }$
C) $y = \frac { 1 } { 36 } x$
D) $y = \frac { 1 } { 6 } x + \frac { 1 } { 3 }$

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 $x = 6$ gives $f'(6) = -\frac{1}{36}$, which is the slope of the tangent line. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with $m = -\frac{1}{36}$ and the point $(6, \frac{1}{6})$, we get $y - \frac{1}{6} = -\frac{1}{36}(x - 6)$. Simplifying this equation gives $y = -\frac{1}{36}x + \frac{1}{3}$, 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 $f ( x ) = \frac { 1 } { x }$