Use the table below to evaluate the derivative with respect to x of g of f of 3 times x at x = 1.
x_____1___2___3___4
f(x) 6 1 2 2
f ′(x) 6 1 10 2
g(x) 1 4 4 3
g ′(x) 4 5 7 –4
d/dx g(f(3x)) = dg/df * df/dx * 3
To evaluate the derivative with respect to x of g of f of 3 times x at x = 1, we need to use the chain rule. The chain rule states that if we have a composite function, the derivative of the composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.
In this case, the composite function is g of f of 3 times x, and the outer function is g(x) while the inner function is f(3x). Let's break down the steps to find the derivative at x = 1 using the chain rule.
1. Evaluate f(3x) at x = 1:
f(3x) = f(3 * 1) = f(3) = 2
2. Evaluate g'(x) at x = 2:
g'(x) = 5
3. Evaluate f'(3x) at x = 1:
f'(3x) = f'(3 * 1) = f'(3) = 10
4. Finally, apply the chain rule to find the derivative of g of f of 3 times x at x = 1:
(gof)'(x) = g'(f(3x)) * f'(3x)
= g'(f(3 * 1)) * f'(3 * 1)
= g'(f(3)) * f'(3)
= g'(2) * 10
= 5 * 10
= 50
Therefore, the derivative with respect to x of g of f of 3 times x at x = 1 is 50.