AP Calculus BC 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple - MCQs - Exam Style Questions
No-Calc Question
Let \(f(x)=b x^{2}+3b x+b^{2}+\dfrac{1}{x^{2}}\) where \(b\neq0\). Which of the following is \(f'(x)\)?
(A) \(2bx+3b-\dfrac{2}{x}\)
(B) \(2bx+3b-\dfrac{2}{x^{3}}\)
(C) \(2bx+3b+b^{2}-\dfrac{2}{x^{2}}\)
(D) \(x^{2}+3x+2b\)
(B) \(2bx+3b-\dfrac{2}{x^{3}}\)
(C) \(2bx+3b+b^{2}-\dfrac{2}{x^{2}}\)
(D) \(x^{2}+3x+2b\)
▶️ Answer/Explanation
Detailed solution
\(\dfrac{d}{dx}(bx^{2})=2bx\), \(\dfrac{d}{dx}(3bx)=3b\), \(\dfrac{d}{dx}(b^{2})=0\), \(\dfrac{d}{dx}(x^{-2})=-2x^{-3}=-\dfrac{2}{x^{3}}\).
So \(f'(x)=2bx+3b-\dfrac{2}{x^{3}}\).
✅ Answer: (B)
▶️ Answer/Explanation
Solution
For \( f(x) = 2x^3 + x^2 – 3 \), the derivative is \( f'(x) = 6x^2 + 2x \).
Evaluate at \( x = 2 \): \( f'(2) = 6(2)^2 + 2(2) = 6 \cdot 4 + 4 = 24 + 4 = 28 \).
✅ Answer: B)