can sumeone plz help me find the derivative of (2x^2 - 12x -2) / ((x^2 + x - 2) ^ 2)
thx a lot
Use the following rule:
derivative of u(x)/v(x) =
[v du/dx - u dv/dx]/ v^2
In your case, du/dx = 4x - 12;
dv/dx = 2 (x^2 + x -2)*(2x +1)
v^2 = (x^2 + x - 2)^4
Put it all together
To find the derivative of (2x^2 - 12x - 2) / ((x^2 + x - 2) ^ 2), you can use the quotient rule.
The quotient rule states that for a function f(x) = u(x) / v(x), the derivative of f(x) is given by:
f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2
In this case, u(x) = 2x^2 - 12x - 2 and v(x) = (x^2 + x - 2)^2.
First, let's find u'(x), which is the derivative of u(x):
u'(x) = d/dx (2x^2 - 12x - 2)
= 4x - 12
Next, let's find v'(x), which is the derivative of v(x):
v'(x) = d/dx [(x^2 + x - 2)^2]
= 2(x^2 + x - 2) * (2x + 1)
= 2(2x^3 + 3x^2 - 2x - 2)
Now, we have u'(x) = 4x - 12, v'(x) = 2(2x^3 + 3x^2 - 2x - 2), u(x) = 2x^2 - 12x - 2, and v(x) = (x^2 + x - 2)^2.
Plugging these values into the quotient rule formula, we get:
f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2
= [(x^2 + x - 2)^2 * (4x - 12) - (2x^2 - 12x - 2) * (2(2x^3 + 3x^2 - 2x - 2))] / [(x^2 + x - 2)^2]^2
Simplifying this expression will give you the derivative of the given function.