Let $f$ be a differentiable function such that $f\left(2\right)=5$ and $f^{\prime }\left(2\right)=3$. What is the value of the derivative of $g\left(x\right)=2f\left(x\right)$ at $x=2$?

• A. $3$
• B. $6$
• C. $10$
• D. $5$

Answer: (B)

Verified step by step guidance

1. Step 1: Recognize that g(x) = 2f(x) is a composite function where g(x) is defined in terms of f(x). To find the derivative of g(x), apply the constant multiple rule of differentiation.
2. Step 2: Recall the constant multiple rule: If g(x) = c * f(x), where c is a constant, then g'(x) = c * f'(x). Here, c = 2.
3. Step 3: Substitute the given values into the derivative formula. Since f'(x) is the derivative of f(x), and f'(2) = 3, we use this value in the formula g'(x) = 2 * f'(x).
4. Step 4: Evaluate g'(2) by substituting x = 2 into the formula. This gives g'(2) = 2 * f'(2).
5. Step 5: Simplify the expression g'(2) = 2 * 3 to find the derivative of g(x) at x = 2. The final value can now be calculated.