Questions: Given (f(1)=1), (f'(1)=7), (g(1)=2), (g'(1)=2), (f'(2)=16), and (g'(2)=11), compute the derivative (left.fracdd x[f(g(x))]rightx=1)

Solution Steps

To find the derivative \(\left.\frac{d}{d x}[f(g(x))]\right|_{x=1}\), we will use the chain rule. The chain rule states that the derivative of a composite function \(f(g(x))\) is \(f'(g(x)) \cdot g'(x)\). We need to evaluate this expression at \(x=1\).

1. Compute \(g(1)\) to find the input for \(f'\).
2. Use the given values to find \(f'(g(1))\) and \(g'(1)\).
3. Multiply these values to find the derivative at \(x=1\).

To find the derivative \(\left.\frac{d}{d x}[f(g(x))]\right|_{x=1}\), we use the chain rule, which states: \[ \frac{d}{d x}[f(g(x))] = f'(g(x)) \cdot g'(x) \]

We need to evaluate this expression at \(x=1\):

• From the problem, we have \(g(1) = 2\), so we need to find \(f'(g(1)) = f'(2)\).
• We are given \(f'(2) = 16\) and \(g'(1) = 2\).

Now we can substitute the values into the derivative: \[ \left.\frac{d}{d x}[f(g(x))]\right|_{x=1} = f'(g(1)) \cdot g'(1) = f'(2) \cdot g'(1) = 16 \cdot 2 = 32 \]

Final Answer