Calculate g'(36^(1/2)), where g is the inverse of f(x) = ((x*x)+16x) for x less than or equal to -16
To find the derivative of g'(36^(1/2)), we need to first find the derivative of the function f(x) and then find its inverse.
Step 1: Find the derivative of f(x)
Given that f(x) = ((x*x) + 16x), we can find its derivative by using the power rule and the sum rule.
Applying the power rule, the derivative of x^2 with respect to x is 2x.
Applying the sum rule, the derivative of 16x with respect to x is 16.
Therefore, the derivative of f(x) is f'(x) = 2x + 16.
Step 2: Find the inverse of f(x) for x ≤ -16
To find the inverse of f(x), we interchange the roles of x and f(x) and solve for x.
Let y = f(x) = ((x*x) + 16x)
Swap x and y:
x = ((y*y) + 16y)
Rearranging the equation:
y^2 + 16y - x = 0
Now, we solve this quadratic equation for y. The inverse of f(x) is denoted as g(x), so we write it as g(x) = y.
Step 3: Calculate g'(36^(1/2))
We need to calculate the derivative of g(x) and substitute x with 36^(1/2).
Differentiating g(x) implicitly with respect to x:
2y * dy/dx + 16 * dy/dx = 0
Rearranging the equation and isolating dy/dx:
dy/dx (2y + 16) = -2y
dy/dx = -2y / (2y + 16)
Since g(x) = y, we replace y with g(x) in the equation:
g'(x) = -2 * g(x) / (2 * g(x) + 16)
Now, substitute x = 36^(1/2) into the equation:
g'(36^(1/2)) = -2 * g(36^(1/2)) / (2 * g(36^(1/2)) + 16)
So, to calculate g'(36^(1/2)), you need to determine the value of g(36^(1/2)) first. Finding the exact value of g(36^(1/2)) would require solving the quadratic equation.