Find the first and second derivative for f(x) = x^2 - 9 / X^2 -4
please show me how you got the answer for both
f = (x^2-9)/(x^2-4)
f = u/v, so f' = (u'v-uv')/v^2
f' = ((2x)(x^2-4) - (x^2-9)(2x))/(x^2-4)^2
= 10x/(x^2-4)^2
or, realize that f(x) = 1 - 5/(x^2-4)
f' = 10x/(x^2-4)^2
f'' = ((10)(x^2-4)^2 - (10x)(2)(x^2-4)(2x))/(x^2-4)^4
= (10(x^2-4) - 40x^2)/(x^2-4)^3
= -10(3x^2+4)/(x^2-4)^3
2x+18x^-3
Use the quotient rule. (DN'-D'N)/(D)^2
D=denominator D'=derivative of denominator
N=numerator N'=derivative of numerator
To find the first and second derivatives of the function f(x) = (x^2 - 9) / (x^2 - 4), we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / h(x)^2
Let's begin by finding the first derivative:
Step 1: Identify the numerator g(x) and the denominator h(x).
g(x) = x^2 - 9
h(x) = x^2 - 4
Step 2: Find the derivatives of the numerator and denominator.
g'(x) = 2x
h'(x) = 2x
Step 3: Apply the quotient rule formula to find the first derivative:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / h(x)^2
= [(2x) * (x^2 - 4) - (x^2 - 9) * (2x)] / (x^2 - 4)^2
Simplifying this expression will give us the first derivative, f'(x).
To find the second derivative, we will differentiate the first derivative, f'(x), using the quotient rule once again. Let's follow the same steps as above:
Step 1: Identify the numerator g(x) and the denominator h(x) of f'(x).
g(x) = (2x) * (x^2 - 4) - (x^2 - 9) * (2x)
h(x) = (x^2 - 4)^2
Step 2: Find the derivatives of the numerator and denominator.
g'(x) = 6x^2 - 24
h'(x) = 2(x^2 - 4)(2x)
Step 3: Apply the quotient rule formula to find the second derivative:
f''(x) = (g'(x) * h(x) - g(x) * h'(x)) / h(x)^2
= [(6x^2 - 24) * (x^2 - 4)^2 - [(2x) * (x^2 - 4) - (x^2 - 9) * (2x)] * 2(x^2 - 4)(2x)] / (x^2 - 4)^4
Simplifying this expression will give us the second derivative, f''(x).
Please note that there may be additional simplifications that can be made.