f'(-2) if f(x)= g(h(x))^3
Chart
x= -3,-2,-1,0,1,2,3
g(x)= 0,1,3,2,0,-2,-3
h(x)= 1,2,0,3,-1,-2,0
g'(x)= 1,2,-1,-2,-2,-1,0
h'(x)= 0,-3,-2,3,-2,0,1
To find f'(-2), we need to find the derivative of f(x) and then evaluate it at x = -2.
Given that f(x) = g(h(x))^3, we can use the chain rule to find the derivative of f(x).
The chain rule states that if we have a composition of functions, like g(h(x)), then the derivative of the composition is given by the derivative of the outer function times the derivative of the inner function.
Let's find the derivative of f(x):
f(x) = g(h(x))^3
Using the chain rule, the derivative of f(x) is:
f'(x) = 3 * g(h(x))^2 * g'(h(x)) * h'(x)
Now, we can substitute the values from the chart into this equation:
During this process, we will evaluate the expressions at each x-value in the chart.
x = -3:
f'(-3) = 3 * g(h(-3))^2 * g'(h(-3)) * h'(-3) = 3 * g(1)^2 * g'(1) * h'(-3) = 3 * 3^2 * 1 * 0 = 0
x = -2:
f'(-2) = 3 * g(h(-2))^2 * g'(h(-2)) * h'(-2) = 3 * g(2)^2 * g'(-2) * h'(-2) = 3 * (-2)^2 * (-1) * (-3) = 36
So, f'(-2) = 36.