find the equation for the tangent line

g(x)=x-1/2x+4 at x=-1

g = x -.5 x + 4

did you mean x^2 maybe? or what?

x-1÷2x+4

g = (x-1)/(2x+4)

dg/dx = [(2x+4)-2(x-1)]/(2x+4)^2

at x = -1
dg/dx = m the slope
= [(-2+4)-2(-2)]/(2)^2
= 6/4 = 3/2
and
g = -2/2 = -1

so
y = m x + b
-1 = 3/2 (-1) + b
b = 1/2
so
y = (3/2) x + 1/2
or 2 y = 3 x + 1

check my arithmetic!

perfect ty! i got y=3/2x+1/2

Good :)

To find the equation for the tangent line to the function g(x) = (x - 1) / (2x + 4) at x = -1, you can follow these steps:

Step 1: Find the derivative of the function g(x) with respect to x, which will give you the slope of the tangent line at any given point.

Step 2: Substitute the value x = -1 into the derivative to obtain the slope of the tangent line at x = -1.

Step 3: Use the slope and the point (x = -1, g(x = -1)) to determine the equation of the tangent line using the point-slope formula.

Let's go through these steps one by one:

Step 1: Find the derivative of g(x):

To find the derivative of g(x), we can use the quotient rule: If f(x) = p(x) / q(x), then f'(x) = (p'(x) * q(x) - q'(x) * p(x)) / (q(x))^2.

In our case, p(x) = (x - 1) and q(x) = (2x + 4).

Differentiating p(x) and q(x) gives us:

p'(x) = 1 (the derivative of x - 1 is 1)
q'(x) = 2 (the derivative of 2x + 4 is 2)

Now we can substitute these values into the quotient rule formula:

g'(x) = [(1 * (2x + 4)) - (2 * (x - 1))] / ((2x + 4)^2)
= (2x + 4 - 2x + 2) / (4x^2 + 16x + 16)
= (6) / (4x^2 + 16x + 16)
= 3 / (2x^2 + 8x + 8)

Step 2: Find the slope of the tangent line at x = -1:

Substitute x = -1 into the derivative g'(x):

g'(-1) = 3 / (2(-1)^2 + 8(-1) + 8)
= 3 / (2 + (-8) + 8)
= 3 / 2

So, the slope of the tangent line at x = -1 is 3/2.

Step 3: Use the point-slope formula to find the equation of the tangent line:

The point-slope formula is given as: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

We have x1 = -1, y1 = g(-1) = (-1 - 1) / (2(-1) + 4) = -2 / 2 = -1.

Substitute these values into the point-slope formula:

y - (-1) = (3/2)(x - (-1))
y + 1 = (3/2)(x + 1)
y + 1 = (3/2)x + 3/2
y = (3/2)x + 3/2 - 1
y = (3/2)x + 3/2 - 2/2
y = (3/2)x + 1/2

Therefore, the equation for the tangent line to g(x) = (x - 1) / (2x + 4) at x = -1 is y = (3/2)x + 1/2.