f(x) and g(x) are a differentiable function for all reals and h(x) = g[f(5x)]. The table below gives selected values for f(x), g(x), f '(x), and g '(x). Find the value of h'(1).
x 1 2 3 4 5 6
f(x) 0 3 2 1 2 0
g(x) 1 3 2 6 5 0
f '(x) 3 2 1 4 0 2
g '(x) 1 5 4 3 2 0
To find the value of h'(1), we need to apply the chain rule, which states that if we have a composite function h(x) = g[f(x)], then the derivative of h(x) with respect to x is h'(x) = g'(f(x)) * f'(x).
In this case, the composite function is h(x) = g[f(5x)], so the derivative of h(x) with respect to x is h'(x) = g'[f(5x)] * f'(5x) * 5.
To find h'(1), we need to substitute x = 1 into the derivative expression and evaluate.
Step 1: Find f(5x) for x = 1.
f(5x) = f(5 * 1) = f(5) = 2 (from the table)
Step 2: Find g'[f(5x)] for f(5x) = 2.
g'[f(5x)] = g'(2) = 4 (from the table)
Step 3: Find f'(5x) for x = 1.
f'(5x) = f'(5 * 1) = f'(5) = 0 (from the table)
Step 4: Substitute the values into h'(x) = g'[f(5x)] * f'(5x) * 5.
h'(1) = 4 * 0 * 5 = 0
Therefore, the value of h'(1) is 0.
dh/dx = dg/df * df/dx * 5
f'(5) = 5*f'(1) = 5*2 = 10
g'(f(5)) = g'(2) = 4
h'(1) = g'(2) * f'(1) * 5 = 4*3*5 = 60