Find the equation of the tangent line to the curve when x has the given value.
-$$f ( x ) = \frac { 4 } { x } ; x = 5$$
A) $y = - \frac { 4 } { 25 } x + \frac { 8 } { 5 }$
B) $y = - 39 x - 80$
C) $y = 13 x - 16$
D) $y = \frac { 1 } { 20 } x + \frac { 1 } { 5 }$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line to the curve at $x = 5$, first, we need to find the derivative of $f(x) = \frac{4}{x}$, which is $f'(x) = -\frac{4}{x^2}$. Evaluating this derivative at $x = 5$ gives $f'(5) = -\frac{4}{25}$, which is the slope of the tangent line. The y-coordinate at $x = 5$ is $f(5) = \frac{4}{5}$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with $m = -\frac{4}{25}$, $x_1 = 5$, and $y_1 = \frac{4}{5}$, we get $y - \frac{4}{5} = -\frac{4}{25}(x - 5)$. Simplifying this equation gives $y = -\frac{4}{25}x + \frac{8}{5}$, which matches option A.

Find the Equation of the Tangent Line to the Curve f(x)=4x;x=5f ( x ) = \frac { 4 } { x } ; x = 5f(x)=x4​;x=5

Find the Equation of the Tangent Line to the Curve $f ( x ) = \frac { 4 } { x } ; x = 5$