# Equation of tangent line to curve

## Homework Statement

Find all x coordinates of points (x, y) on the curve y = (x − 2)5 /(x − 4)3 where the tangent line is horizontal

## Homework Equations

Quotient Rule - Differentiation

## The Attempt at a Solution

So I could guess, that if tangent line is horizontal, the equation of tangent line to curve is y = c, where gradient is 0.

So, first derivative of the curve is suppose equal to zero.

dy/dx;
0 = ((5)(x - 2)4(1)(x - 4)3 - (x - 2)5(3)(x - 4)2(1)) / ((x - 4)3)2
0 = (5(x - 2)4(x - 4)3 - 3(x - 2)5(x - 4)2) / (x - 4)6

Okay now I'm stuck.

Mark44
Mentor

## Homework Statement

Find all x coordinates of points (x, y) on the curve y = (x − 2)5 /(x − 4)3 where the tangent line is horizontal

## Homework Equations

Quotient Rule - Differentiation

## The Attempt at a Solution

So I could guess, that if tangent line is horizontal, the equation of tangent line to curve is y = c, where gradient is 0.

So, first derivative of the curve is suppose equal to zero.

dy/dx;
0 = ((5)(x - 2)4(1)(x - 4)3 - (x - 2)5(3)(x - 4)2(1)) / ((x - 4)3)2
0 = (5(x - 2)4(x - 4)3 - 3(x - 2)5(x - 4)2) / (x - 4)6

Okay now I'm stuck.
That's a good start.

What you need to do is to write the numerator in a factored form. The best way to do that is to find the common factors in 5(x - 2)4(x - 4)3 - 3(x - 2)5(x - 4)2.

The result will look like this: (x - 2)m(x - 4)n(<what's left>)

So it would be like this,
((x - 2)4(x - 4)2(5(x - 4) - 3(x - 2))) / (x - 4)6
( (x - 2)4(5(x - 4) - 3(x - 2)) ) / (x - 4)4

Now I stuck again.

Last edited:
Dick