How do you find the equation of the tangent line to the graph of the given function #f(x)= x/(x-2)#; at (3,3)?

Question

How do you find the equation of the tangent line to the graph of the given function  #f(x)= x/(x-2)#; at (3,3)?

Christopher Allers · Accepted Answer

#y = -2x + 9#

To find the equation of the line tangent to the graph of the function at a given point, in your case#(3,3)#, you need to calculate the first derivative of the function first.

The first derivative of the function will give you the slope of this line

#color(blue)("slope" = m = d/dxf(x))#

Use the quotient rule to differentiate the function

#d/dxf(x) = ([d/dx(x)] * (x-2) - x * d/dx(x-2))/(x-2)^2#

#f^'(x) = [1 * (x-2) - x * 1]/(x-2)^2#

#f^'(x) = (color(red)(cancel(color(black)(x))) - 2 - color(red)(cancel(color(black)(x))))/(x-2)^2 = -2/(x-2)^2#

The slope of the tangent line at point #(3,3)# will be

#m = f^'(3) = -2/(3-2)^2 = -2/1 = -2#

The equation of the line in point slope form will be

#color(blue)(y - y_1 = m * (x - x_1))#

#y - 3 = -2 * (x-3)#

#y = -2x + 6 + 3#

#y = -2x + 9#

Axel Alvarez · Answer

undefined To find the equation of the tangent line to the graph of the function f(x) = x/(x-2) at the point (3,3), we can follow these steps:

1. Find the derivative of the function f(x) using the quotient rule.
2. Substitute the x-coordinate of the given point (3,3) into the derivative to find the slope of the tangent line.
3. Use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope, to write the equation of the tangent line.

Let's go through these steps:

1. The derivative of f(x) = x/(x-2) can be found using the quotient rule:
   f'(x) = [(x-2)(1) - x(1)] / (x-2)^2

2. Substitute x = 3 into the derivative to find the slope of the tangent line:
   f'(3) = [(3-2)(1) - 3(1)] / (3-2)^2

3. Simplify the expression:
   f'(3) = [1 - 3] / 1^2
   f'(3) = -2

4. Now we have the slope of the tangent line, which is -2, and the point (3,3). We can use the point-slope form to write the equation of the tangent line:
   y - 3 = -2(x - 3)

5. Simplify the equation:
   y - 3 = -2x + 6

6. Rearrange the equation to the standard form:
   2x + y = 9

Therefore, the equation of the tangent line to the graph of f(x) = x/(x-2) at the point (3,3) is 2x + y = 9.