Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


g. ƒ(x + g(x)), x = 0

Verified step by step guidance

1. To find the derivative of the composite function \( h(x) = f(x + g(x)) \), we will use the chain rule. The chain rule states that if you have a composite function \( h(x) = f(u(x)) \), then the derivative \( h'(x) = f'(u(x)) \cdot u'(x) \).
2. First, identify the inner function \( u(x) = x + g(x) \). The derivative of \( u(x) \) with respect to \( x \) is \( u'(x) = 1 + g'(x) \).
3. Next, evaluate \( u(x) \) at \( x = 0 \). Since \( g(0) = 1 \), we have \( u(0) = 0 + 1 = 1 \).
4. Now, find \( f'(u(x)) \) at \( u(0) = 1 \). From the given data, \( f'(1) = \frac{1}{2} \).
5. Finally, apply the chain rule: \( h'(0) = f'(u(0)) \cdot u'(0) = f'(1) \cdot (1 + g'(0)) = \frac{1}{2} \cdot (1 + \frac{1}{2}) \).