Question

Given $F(x)=(f(g(x)))^{2}$, $g(1)=2$, $g'(1)=3$, $f(2)=4$ and $f'(2)=5$. Then the value of $F'(1)$ is equal to

• A. 25
• B. 100
• C. 75
• D. 50
• E. 120

Hint:

Calculus Tip: When using the Chain Rule on multiple nested functions like $y = [f(g(x))]^n$, always work strictly from the "outside in": Power first, then outer function $f^{\prime}$, then inner function $g^{\prime}$.

Answer 1

Answer: (E)

Concept:
To find the derivative of a composite function raised to a power, apply the Chain Rule multiple times. The outer layer is the power rule, the middle layer is the function $f$, and the inner layer is the function $g$.

Step 1: Apply the general power rule.
We are given $F(x) = (f(g(x)))^2$. Differentiating the outer square gives: $$F^{\prime}(x) = 2(f(g(x)))^{1} \cdot \frac{d}{dx}[f(g(x))]$$

Step 2: Apply the Chain Rule to the inner function.
The derivative of $f(g(x))$ is $f^{\prime}(g(x)) \cdot g^{\prime}(x)$. Substitute this back: $$F^{\prime}(x) = 2f(g(x)) \cdot f^{\prime}(g(x)) \cdot g^{\prime}(x)$$

Step 3: Substitute the target value $x=1$.
We need to evaluate the derivative at $x=1$: $$F^{\prime}(1) = 2f(g(1)) \cdot f^{\prime}(g(1)) \cdot g^{\prime}(1)$$

Step 4: Evaluate the inner values from the given data.
We are given $g(1) = 2$ and $g^{\prime}(1) = 3$. Substitute these in: $$F^{\prime}(1) = 2f(2) \cdot f^{\prime}(2) \cdot (3)$$

Step 5: Perform the final calculation.
We are also given $f(2) = 4$ and $f^{\prime}(2) = 5$. Substitute these final values: $$F^{\prime}(1) = 2(4) \cdot (5) \cdot 3$$ $$F^{\prime}(1) = 8 \cdot 15 = 120$$ Hence the correct answer is (E) 120.