if y = 2f(g(x)), then d^2y / dx^2 =
A. f''(g(x)) * g'(x)
B. f''(g(x)) * [g'(x)]^2
C. 2f''(g(x)) * [g'(x)]^2
D. 2f''(g(x)) * [g'(x)]^2 + 2f'(g(x)) * g''(x)
E. Cannot be determined
use the chain rule
y' = 2f'(g)*g'(x)
now use the product rule
y" = 2f"(g)g'(x)*g'(x) + 2f'(g)g"(x)
oobleck is correct; answer d is the right answer!
To find the second derivative of y with respect to x, we need to apply the chain rule twice.
First, let's find the first derivative of y with respect to x:
dy/dx = [d(2f(g(x)))/d(g(x))] * [d(g(x))/dx]
Applying the chain rule, we have:
dy/dx = 2f'(g(x)) * g'(x)
Now, let's find the second derivative of y with respect to x:
d^2y/dx^2 = [d(2f'(g(x)) * g'(x))/d(g(x))] * [d(g(x))/dx]
Applying the chain rule again, we have:
d^2y/dx^2 = [2f''(g(x)) * g'(x) + 2f'(g(x)) * g''(x)] * [d(g(x))/dx]
However, we cannot determine the exact value of [d(g(x))/dx] without more information. Therefore, the correct answer is:
E. Cannot be determined
To find the second derivative of y with respect to x, we can use the chain rule. Recall that the chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
In this case, y = 2f(g(x)), so let's differentiate both sides with respect to x:
dy/dx = d/dx (2f(g(x)))
Using the chain rule, the right-hand side can be written as 2 * f'(g(x)) * g'(x). This gives us the first derivative of y with respect to x.
Now, to find the second derivative, we need to differentiate dy/dx with respect to x:
d^2y/dx^2 = d/dx (2 * f'(g(x)) * g'(x))
Using the product rule, the derivative of 2 * f'(g(x)) * g'(x) with respect to x can be expanded as follows:
d^2y/dx^2 = 2 * [f''(g(x)) * g'(x) + f'(g(x)) * g''(x)]
Therefore, the correct answer is D. 2f''(g(x)) * [g'(x)]^2 + 2f'(g(x)) * g''(x).