Question

If \( y = f(x^2 + 2) \) and \( f'(3) = 5 \), then \( \frac{dy}{dx} \) at \( x = 1 \) is:

• A. \( 5 \)
• B. \( 25 \)
• C. \( 15 \)
• D. \( 20 \)
• E. \( 10 \)

Hint:

Always identify the "inner" function first. A common mistake is forgetting to multiply by the derivative of that inner part (\(2x\) in this case).

Answer 1

Answer: (E)

Understanding the Concept: To find the derivative of a composite function, we use the Chain Rule. If \( y = f(g(x)) \), then: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \] This allows us to differentiate the outer function while keeping the inner function intact, then multiply by the derivative of that inner function.

Step 1: Differentiate \( y \) with respect to \( x \).
Given \( y = f(x^2 + 2) \). Let \( u = x^2 + 2 \). Applying the Chain Rule: \[ \frac{dy}{dx} = f'(x^2 + 2) \cdot \frac{d}{dx}(x^2 + 2) \] \[ \frac{dy}{dx} = f'(x^2 + 2) \cdot (2x) \]

Step 2: Substitute \( x = 1 \) into the derivative.
\[ \left. \frac{dy}{dx} \right|_{x=1} = f'(1^2 + 2) \cdot (2 \cdot 1) \] \[ \left. \frac{dy}{dx} \right|_{x=1} = f'(3) \cdot 2 \]

Step 3: Plug in the given value of \( f'(3) \).
We are given \( f'(3) = 5 \). \[ \left. \frac{dy}{dx} \right|_{x=1} = 5 \cdot 2 = 10 \]