What is the equation of the line tangent to f(x)=(-3x-1)/(x+4) at x=-2 ?

1 Answer

Tangent Line is

11x+4y=-12

Explanation:

Given equation f(x)=(-3x-1)/(x+4) at the point x=-2

Let us solve the point (x_1, y_1)
Let x_1=-2

Using y=(-3x-1)/(x+4)

y_1=(-3x_1-1)/(x_1+4)

y_1=(-3(-2)-1)/(-2+4)

y_1=5/2

We have (x_1, y_1)=(-2, 5/2)

Let us solve the slope m

m=f' (x) at x=x_1=-2

m=d/dx((-3x-1)/(x+4))

m=((x+4)*d/dx(-3x-1)-(-3x-1)*d/dx(x+4))/(x+4)^2

m=((x+4)*(-3)-(-3x-1)*1)/(x+4)^2

m=((-2+4)*(-3)-(-3(-2)-1)*1)/(-2+4)^2

m=((2)*(-3)-(6-1)*1)/(2)^2

m=-11/4

Let us solve the tangent line

y-y_1=m(x-x_1)

y-5/2=(-11)/4(x--2)

y-5/2=(-11)/4(x+2)

4y-10=(-11)(x+2)

4y=(-11)(x+2)+10

4y=-11x-22+10

11x+4y=-12

Kindly see the graph of f(x)=(-3x-1)/(x+4) and 11x+4y=-12

enter image source here

God bless....I hope the explanation is useful.