Unit 2.7 – Derivatives of cos x, sin x, e^x, and ln x

AP® Calculus AB & BC | Essential Transcendental Function Derivatives

Core Concept: Up until now, you've learned to differentiate polynomials and power functions. Topic 2.7 introduces the derivatives of four transcendental functions—functions that go beyond algebraic operations! These four formulas: sin x, cos x, e^x, and ln x are absolutely fundamental and must be memorized cold. They appear EVERYWHERE in calculus and form the building blocks for differentiating all trigonometric, exponential, and logarithmic functions. Master these four derivatives—they're non-negotiable for the AP® exam!

🌟 The Big Four: Must-Memorize Derivatives

MEMORIZE THESE FOUR FORMULAS!

Function	Derivative	Memory Aid
\(\sin x\)	\(\cos x\)	Sine → Cosine (co-function)
\(\cos x\)	\(-\sin x\)	Cosine → Negative Sine ⚠️
\(e^x\)	\(e^x\)	e stays the same! ✨
\(\ln x\)	\(\frac{1}{x}\)	Natural log → Reciprocal

These four derivatives are the foundation of all transcendental function differentiation!

📐 Detailed Breakdown: Each Derivative Explained

DERIVATIVE 1: SINE FUNCTION

\[ \frac{d}{dx}[\sin x] = \cos x \]

In Words: The derivative of sine is cosine.

Why This Makes Sense:

Sine's rate of change is highest when sine = 0 (at \(x = 0, \pm\pi, \pm 2\pi\)...)
At these points, cosine = ±1 (maximum rate of change)
When sine is at its peak (sine = 1), the rate of change is zero, and cosine = 0

Proof Sketch (From Limit Definition):

\[ \frac{d}{dx}[\sin x] = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h} \]

Using the angle addition formula \(\sin(x+h) = \sin x \cos h + \cos x \sin h\) and the special limits:

\(\lim_{h \to 0} \frac{\sin h}{h} = 1\)
\(\lim_{h \to 0} \frac{\cos h - 1}{h} = 0\)

We get \(\frac{d}{dx}[\sin x] = \cos x\) ✓

Examples:

\(\frac{d}{dx}[\sin x] = \cos x\)
\(\frac{d}{dx}[5\sin x] = 5\cos x\)
\(\frac{d}{dx}[\sin x + x^2] = \cos x + 2x\)

DERIVATIVE 2: COSINE FUNCTION

\[ \frac{d}{dx}[\cos x] = -\sin x \]

In Words: The derivative of cosine is negative sine.

⚠️ CRITICAL: Don't forget the negative sign! This is the #1 mistake students make.

Why the Negative Sign?

At \(x = 0\), cosine is at its maximum (cos 0 = 1)
As x increases, cosine decreases → negative rate of change
The slope at \(x = 0\) is 0 (sine = 0), but just after, it's negative
The cosine curve is "going down" to the right of \(x = 0\)

Proof Sketch:

\[ \frac{d}{dx}[\cos x] = \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h} \]

Using \(\cos(x+h) = \cos x \cos h - \sin x \sin h\) and the same special limits, we get \(-\sin x\) ✓

Examples:

\(\frac{d}{dx}[\cos x] = -\sin x\)
\(\frac{d}{dx}[3\cos x] = -3\sin x\)
\(\frac{d}{dx}[\cos x - \sin x] = -\sin x - \cos x\)
\(\frac{d}{dx}[-\cos x] = -(-\sin x) = \sin x\) (double negative!)

DERIVATIVE 3: NATURAL EXPONENTIAL FUNCTION

\[ \frac{d}{dx}[e^x] = e^x \]

In Words: The derivative of \(e^x\) is itself! This is the most special property in all of calculus.

Why e is Special:

\(e \approx 2.71828...\) is the only base where \(\frac{d}{dx}[a^x] = a^x\)
For any other base: \(\frac{d}{dx}[a^x] = a^x \ln a\) (extra constant!)
This is WHY we use e as the "natural" exponential base
The function \(e^x\) grows at a rate proportional to its current value

Proof Sketch:

\[ \frac{d}{dx}[e^x] = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h} = e^x \lim_{h \to 0} \frac{e^h - 1}{h} \]

The special limit \(\lim_{h \to 0} \frac{e^h - 1}{h} = 1\) is essentially the definition of e!

Examples:

\(\frac{d}{dx}[e^x] = e^x\)
\(\frac{d}{dx}[7e^x] = 7e^x\)
\(\frac{d}{dx}[e^x + x^3] = e^x + 3x^2\)
\(\frac{d}{dx}[-e^x] = -e^x\)

For Other Bases (General Exponential):

\[ \frac{d}{dx}[a^x] = a^x \ln a \]

Example: \(\frac{d}{dx}[2^x] = 2^x \ln 2\)

DERIVATIVE 4: NATURAL LOGARITHM FUNCTION

\[ \frac{d}{dx}[\ln x] = \frac{1}{x} \quad (x > 0) \]

In Words: The derivative of natural log is one over x (reciprocal of x).

Domain Note: \(\ln x\) is only defined for \(x > 0\), so the derivative only exists for positive x.

Key Properties:

As \(x \to \infty\), \(\frac{1}{x} \to 0\) (derivative approaches 0—ln grows slowly)
As \(x \to 0^+\), \(\frac{1}{x} \to \infty\) (derivative blows up—steep near origin)
The derivative is always positive for \(x > 0\) (ln is always increasing)

Proof Sketch:

Since \(e^{\ln x} = x\), differentiate both sides using the chain rule:

\[ e^{\ln x} \cdot \frac{d}{dx}[\ln x] = 1 \] \[ x \cdot \frac{d}{dx}[\ln x] = 1 \] \[ \frac{d}{dx}[\ln x] = \frac{1}{x} \quad \checkmark \]

Examples:

\(\frac{d}{dx}[\ln x] = \frac{1}{x}\)
\(\frac{d}{dx}[2\ln x] = \frac{2}{x}\)
\(\frac{d}{dx}[\ln x + x] = \frac{1}{x} + 1\)

For Other Bases (General Logarithm):

\[ \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a} \]

Example: \(\frac{d}{dx}[\log_{10} x] = \frac{1}{x \ln 10}\)

📋 Quick Reference: All Four Derivatives

Function f(x)	Derivative f'(x)	Key Point
\(\sin x\)	\(\cos x\)	Sine → Cosine
\(\cos x\)	\(-\sin x\)	Negative sign!
\(e^x\)	\(e^x\)	Derivative = itself
\(\ln x\)	\(\frac{1}{x}\)	Reciprocal of x

📖 Comprehensive Worked Examples

Example 1: Basic Trig Combination

Problem: Find \(f'(x)\) for \(f(x) = 3\sin x - 2\cos x\)

Solution:

Apply sum/difference rule: Differentiate term-by-term
First term: \(\frac{d}{dx}[3\sin x] = 3\cos x\)
Second term: \(\frac{d}{dx}[-2\cos x] = -2(-\sin x) = 2\sin x\)
Combine: \(f'(x) = 3\cos x + 2\sin x\)

Answer: \(f'(x) = 3\cos x + 2\sin x\)

Example 2: Exponential and Polynomial

Problem: Differentiate \(g(x) = 5e^x + 2x^3 - 7\)

Solution:

Term 1: \(\frac{d}{dx}[5e^x] = 5e^x\) (exponential stays same!)
Term 2: \(\frac{d}{dx}[2x^3] = 6x^2\) (power rule)
Term 3: \(\frac{d}{dx}[-7] = 0\) (constant rule)
Combine: \(g'(x) = 5e^x + 6x^2\)

Answer: \(g'(x) = 5e^x + 6x^2\)

Example 3: Natural Logarithm with Polynomial

Problem: Find \(\frac{dy}{dx}\) for \(y = 4\ln x + x^2 - \frac{1}{x}\)

Solution:

Rewrite: \(y = 4\ln x + x^2 - x^{-1}\)
Term 1: \(\frac{d}{dx}[4\ln x] = \frac{4}{x}\)
Term 2: \(\frac{d}{dx}[x^2] = 2x\)
Term 3: \(\frac{d}{dx}[-x^{-1}] = -(-1)x^{-2} = x^{-2} = \frac{1}{x^2}\)
Combine: \(\frac{dy}{dx} = \frac{4}{x} + 2x + \frac{1}{x^2}\)

Answer: \(\frac{dy}{dx} = \frac{4}{x} + 2x + \frac{1}{x^2}\)

Example 4: All Four Functions Together

Problem: Find \(h'(x)\) for \(h(x) = \sin x + \cos x + e^x + \ln x\)

Solution:

Differentiate each term:
- \(\frac{d}{dx}[\sin x] = \cos x\)
- \(\frac{d}{dx}[\cos x] = -\sin x\)
- \(\frac{d}{dx}[e^x] = e^x\)
- \(\frac{d}{dx}[\ln x] = \frac{1}{x}\)
Combine: \(h'(x) = \cos x - \sin x + e^x + \frac{1}{x}\)

Answer: \(h'(x) = \cos x - \sin x + e^x + \frac{1}{x}\)

Example 5: Evaluating at a Specific Point

Problem: If \(f(x) = 2\sin x + e^x\), find \(f'(0)\)

Solution:

Find derivative: \(f'(x) = 2\cos x + e^x\)
Evaluate at x = 0:
- \(f'(0) = 2\cos(0) + e^0\)
- \(= 2(1) + 1\) (remember: \(\cos 0 = 1\) and \(e^0 = 1\))
- \(= 3\)

Answer: \(f'(0) = 3\)

Example 6: Mixed with Fractions

Problem: Differentiate \(k(x) = \frac{\sin x}{2} + \frac{e^x}{3} - \frac{\ln x}{4}\)

Solution:

Rewrite: \(k(x) = \frac{1}{2}\sin x + \frac{1}{3}e^x - \frac{1}{4}\ln x\)
Differentiate term-by-term:
\[ k'(x) = \frac{1}{2}\cos x + \frac{1}{3}e^x - \frac{1}{4} \cdot \frac{1}{x} \]
Simplify: \(k'(x) = \frac{\cos x}{2} + \frac{e^x}{3} - \frac{1}{4x}\)

Answer: \(k'(x) = \frac{\cos x}{2} + \frac{e^x}{3} - \frac{1}{4x}\)

🔢 General Formulas: Other Bases

General Exponential Function (base a)

\[ \frac{d}{dx}[a^x] = a^x \ln a \quad (a > 0, a \neq 1) \]

Examples:

\(\frac{d}{dx}[2^x] = 2^x \ln 2\)
\(\frac{d}{dx}[10^x] = 10^x \ln 10\)
\(\frac{d}{dx}[3^x] = 3^x \ln 3\)

Note: When \(a = e\), \(\ln e = 1\), so \(\frac{d}{dx}[e^x] = e^x \cdot 1 = e^x\) ✓

General Logarithm Function (base a)

\[ \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a} \quad (a > 0, a \neq 1, x > 0) \]

Examples:

\(\frac{d}{dx}[\log_{10} x] = \frac{1}{x \ln 10}\)
\(\frac{d}{dx}[\log_2 x] = \frac{1}{x \ln 2}\)

Note: When \(a = e\), \(\ln e = 1\), so \(\frac{d}{dx}[\ln x] = \frac{1}{x \cdot 1} = \frac{1}{x}\) ✓

🎯 Important Limits (Used in Proofs)

📝 Essential Limits: These limits are fundamental to proving the derivative formulas:

Sine Limit:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
Cosine Limit:
\[ \lim_{x \to 0} \frac{\cos x - 1}{x} = 0 \]
Exponential Limit:
\[ \lim_{h \to 0} \frac{e^h - 1}{h} = 1 \]

These limits come from geometry (trig) and the definition of e (exponential). They're tested on the AP® exam!

💡 Tips, Tricks & Memory Aids

✅ Essential Tips

Memorize cold: These four derivatives must be instant recall—no hesitation!
Negative sign alert: Cosine is the ONLY one with a negative sign in its derivative
e is special: Only \(e^x\) has derivative equal to itself
Log domain: \(\ln x\) only defined for \(x > 0\), so derivative too
Practice evaluating: Know special values like \(\sin 0, \cos 0, e^0, \ln 1\)
Combine with other rules: Use these WITH power rule, sum rule, etc.

🔥 Memory Devices

Mnemonic for Trig Derivatives:

"Sine goes to Cosine (co-function),
Cosine goes Negative (that's the key!)."

Pattern for Repeated Differentiation:

If you keep differentiating sine or cosine, you get a 4-step cycle:

\[ \sin x \to \cos x \to -\sin x \to -\cos x \to \sin x \to ... \]

\[ \cos x \to -\sin x \to -\cos x \to \sin x \to \cos x \to ... \]

For Exponential and Log:

"e to the x stays intact,
ln x gives one over x back!"

🎯 Special Values to Memorize

Trig Values:

\(\sin 0 = 0\), \(\cos 0 = 1\)
\(\sin(\pi/2) = 1\), \(\cos(\pi/2) = 0\)
\(\sin \pi = 0\), \(\cos \pi = -1\)

Exponential/Log Values:

\(e^0 = 1\)
\(e^1 = e \approx 2.718\)
\(\ln 1 = 0\)
\(\ln e = 1\)

❌ Common Mistakes to Avoid

Mistake 1: Forgetting the negative sign: \(\frac{d}{dx}[\cos x] = \sin x\) ❌ → It's \(-\sin x\) ✓
Mistake 2: Thinking \(\frac{d}{dx}[e^x] = xe^{x-1}\) ❌ → It's just \(e^x\) ✓
Mistake 3: Writing \(\frac{d}{dx}[\ln x] = \frac{1}{\ln x}\) ❌ → It's \(\frac{1}{x}\) ✓
Mistake 4: Using these formulas with chain rule problems (need Topic 2.9!)
Mistake 5: Forgetting domain: \(\ln x\) derivative only valid for \(x > 0\)
Mistake 6: Mixing up \(\log_{10} x\) and \(\ln x\) (different bases!)
Mistake 7: Double negative confusion: \(\frac{d}{dx}[-\cos x] = -(-\sin x) = \sin x\) ✓
Mistake 8: Not using radians (trig derivatives require radian mode!)

📝 Practice Problems

Find the derivative of each function:

\(f(x) = 4\sin x + 3\cos x\)
\(g(x) = e^x - 2x^2 + 5\)
\(h(x) = \ln x + \frac{1}{x} + \sqrt{x}\)
\(k(x) = -\cos x + \sin x + e^x\)
\(f(x) = 2^x + \log_2 x\) (use general formulas)
If \(f(x) = \sin x + \cos x\), find \(f'(\pi/2)\)

Answers:

\(f'(x) = 4\cos x - 3\sin x\)
\(g'(x) = e^x - 4x\)
\(h'(x) = \frac{1}{x} - \frac{1}{x^2} + \frac{1}{2\sqrt{x}}\) or \(\frac{1}{x} - x^{-2} + \frac{1}{2}x^{-1/2}\)
\(k'(x) = \sin x + \cos x + e^x\) (watch double negative!)
\(f'(x) = 2^x \ln 2 + \frac{1}{x \ln 2}\)
\(f'(x) = \cos x - \sin x\), so \(f'(\pi/2) = \cos(\pi/2) - \sin(\pi/2) = 0 - 1 = -1\)

✏️ AP® Exam Success Tips

What the AP® Exam Expects:

Instant recall: No time to derive—know all four derivatives immediately
Correct notation: Use proper notation: \(\frac{d}{dx}\), \(f'(x)\), etc.
Show work on FRQ: Write out each differentiation step
Evaluate at specific points: Often asked to find \(f'(a)\) for given a
Combine with other rules: Mix these with power rule, sum rule, etc.
Know special limits: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) appears on exams
Use radians: All trig must be in radian mode on calculator
Domain awareness: Note when \(x > 0\) for ln functions

Common FRQ Formats:

"Find f'(x) for the function..." (direct differentiation)
"Find the slope of the tangent line at x = a" (evaluate derivative)
"Find an equation of the line tangent to f at x = a" (use f'(a) as slope)
"Evaluate \(\lim_{x \to 0} \frac{\sin x}{x}\)" (know special limits)
"Show that f'(x) = ..." (prove derivative equals given expression)
"Find the rate of change..." (real-world context requiring derivatives)
"For what values of x is f'(x) = 0?" (critical points)

⚡ Ultimate Quick Reference

THE BIG FOUR - MEMORIZE NOW!

\(\frac{d}{dx}[\sin x] = \cos x\)	\(\frac{d}{dx}[\cos x] = -\sin x\) ⚠️
\(\frac{d}{dx}[e^x] = e^x\) ✨	\(\frac{d}{dx}[\ln x] = \frac{1}{x}\)

Bonus: \(\frac{d}{dx}[a^x] = a^x \ln a\) | \(\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}\)

Special Limits: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) | \(\lim_{x \to 0} \frac{e^x - 1}{x} = 1\)

🔗 Why This Topic Matters

Topic 2.7 connects to:

Topic 2.8: Product and quotient rules use these basic derivatives
Topic 2.9: Chain rule extends these to composite functions (\(\sin(3x), e^{x^2}\), etc.)
Unit 3: Optimization of trig/exponential functions requires these derivatives
Unit 4: Motion problems with harmonic motion use trig derivatives
Unit 5: Integration reverses these (antiderivatives)
Unit 6: Exponential growth/decay models use \(e^x\) derivative
BC Only - Unit 10: Taylor series use repeated differentiation of these functions

Remember: Topic 2.7 introduces the four must-memorize derivatives of transcendental functions: (1) \(\frac{d}{dx}[\sin x] = \cos x\)—sine becomes cosine; (2) \(\frac{d}{dx}[\cos x] = -\sin x\)—cosine becomes negative sine (don't forget that minus sign!); (3) \(\frac{d}{dx}[e^x] = e^x\)—the exponential function is its own derivative (unique to base e); (4) \(\frac{d}{dx}[\ln x] = \frac{1}{x}\)—natural log gives reciprocal. These four formulas, combined with the power rule and linearity rules, form the complete toolkit for basic differentiation. For other bases: \(\frac{d}{dx}[a^x] = a^x \ln a\) and \(\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}\). Master these NOW—they're tested constantly on the AP® exam and used in EVERY subsequent topic! 🎯✨

Unit 2.7 – Derivatives of cos x, sin x, ex, and ln x

🌟 The Big Four: Must-Memorize Derivatives

📐 Detailed Breakdown: Each Derivative Explained

📋 Quick Reference: All Four Derivatives

📖 Comprehensive Worked Examples

Example 1: Basic Trig Combination

Example 2: Exponential and Polynomial

Example 3: Natural Logarithm with Polynomial

Example 4: All Four Functions Together

Example 5: Evaluating at a Specific Point

Example 6: Mixed with Fractions

🔢 General Formulas: Other Bases

🎯 Important Limits (Used in Proofs)

💡 Tips, Tricks & Memory Aids

✅ Essential Tips

🔥 Memory Devices

🎯 Special Values to Memorize

❌ Common Mistakes to Avoid

📝 Practice Problems

✏️ AP® Exam Success Tips

⚡ Ultimate Quick Reference

🔗 Why This Topic Matters

Unit 2.7 – Derivatives of cos x, sin x, e^x, and ln x