AP Calculus AB 2.7 Derivatives of cos x, sin x, ex, and ln x - MCQs - Exam Style Questions

\(\ \displaystyle \frac{d}{dx}\left(x^{5}-5^{x}\right)=\)

(A) \(\dfrac{x^{6}}{6}-\dfrac{5^{x}}{\ln 5}\)
(B) \(5x^{4}-5^{x}\)
(C) \(5x^{4}-x\cdot 5^{x-1}\)
(D) \(5x^{4}-(\ln 5)\,5^{x}\)

▶️ Answer/Explanation

\(\dfrac{d}{dx}(x^{5})=5x^{4}\).
\(\dfrac{d}{dx}(5^{x})=(\ln 5)\,5^{x}\).
Combine with minus sign: \(5x^{4}-(\ln 5)\,5^{x}\).

✅ Answer: (D)

If \(y=\cos x-\ln(2x)\), then \(\dfrac{d^{3}y}{dx^{3}}=\) ?

(A) \(\sin x-\dfrac{2}{x^{3}}\)
(B) \(-\sin x-\dfrac{2}{x^{3}}\)
(C) \(\sin x-\dfrac{1}{x^{3}}\)
(D) \(-\sin x-\dfrac{1}{x^{3}}\)

▶️ Answer/Explanation

\(y’=-\sin x-\dfrac{1}{x}\).
\(y”=-\cos x+\dfrac{1}{x^{2}}\).
\(y”’=\sin x-\dfrac{2}{x^{3}}\).
✅ Answer: (A)

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