AP Calculus BC 2.7 Derivatives of cos x, sin x, ex, and ln x Study Notes - New Syllabus
AP Calculus BC 2.7 Derivatives of cos x, sin x, ex, and ln x Study Notes- New syllabus
AP Calculus BC 2.7 Derivatives of cos x, sin x, ex, and ln x Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Calculate derivatives of familiar functions.
Key Concepts:
- Derivatives of basic functions— $\cos x$, $\sin x$, $e^x$, and $\ln x$
Derivatives of basic functions— $\cos x$, $\sin x$, $e^x$, and $\ln x$
Derivative of \( \sin x \)
The derivative of the sine function is the cosine function.
Formula: If \( f(x) = \sin x \), then \( f'(x) = \cos x \)
Example:
Differentiate \( f(x) = \sin x \)
▶️Answer/Explanation
\( f'(x) = \cos x \)
Example:
Differentiate \( f(x) = 3\sin x \)
▶️Answer/Explanation
\( f'(x) = 3\cos x \)
Derivative of \( \cos x \)
The derivative of cosine is the negative sine function.
Formula: If \( f(x) = \cos x \), then \( f'(x) = -\sin x \)
Example:
Differentiate \( f(x) = \cos x \)
▶️Answer/Explanation
\( f'(x) = -\sin x \)
Example:
Differentiate \( f(x) = -2\cos x \)
▶️Answer/Explanation
\( f'(x) = -2(-\sin x) = 2\sin x \)
Derivative of \( e^x \)
The exponential function \( e^x \) is special — its derivative is itself.
Formula: If \( f(x) = e^x \), then \( f'(x) = e^x \)
Example:
Differentiate \( f(x) = e^x \)
▶️Answer/Explanation
\( f'(x) = e^x \)
Example:
Differentiate \( f(x) = 5e^x \)
▶️Answer/Explanation
\( f'(x) = 5e^x \)
Derivative of \( \ln x \)
The natural logarithmic function’s derivative is the reciprocal of \( x \).
Formula: If \( f(x) = \ln x \), then \( f'(x) = \dfrac{1}{x} \), for \( x > 0 \)
Example:
Differentiate \( f(x) = \ln x \)
▶️Answer/Explanation
\( f'(x) = \dfrac{1}{x} \)
Example:
Differentiate \( f(x) = 4\ln x \)
▶️Answer/Explanation
\( f'(x) = 4 \cdot \dfrac{1}{x} = \dfrac{4}{x} \)
Example:
Differentiate the function:
\( f(x) = 3\sin x + 2e^x – 4\ln x + 5\cos x \)
▶️Answer/Explanation
- \( \frac{d}{dx}(3\sin x) = 3\cos x \)
- \( \frac{d}{dx}(2e^x) = 2e^x \)
- \( \frac{d}{dx}(-4\ln x) = -\frac{4}{x} \)
- \( \frac{d}{dx}(5\cos x) = 5(-\sin x) = -5\sin x \)
Final Answer:
\( f'(x) = 3\cos x + 2e^x – \frac{4}{x} – 5\sin x \)
| Rule | Formula |
|---|---|
| Derivative of \( \sin x \) | \( \frac{d}{dx}(\sin x) = \cos x \) |
| Derivative of \( \cos x \) | \( \frac{d}{dx}(\cos x) = -\sin x \) |
| Derivative of \( e^x \) | \( \frac{d}{dx}(e^x) = e^x \) |
| Derivative of \( \ln x \) | \( \frac{d}{dx}(\ln x) = \frac{1}{x} \), for \( x > 0 \) |