Derivatives of
Trig & Log Functions
Master the derivatives of \(\sin x\), \(\cos x\), \(e^x\), and \(\ln x\): the four building blocks that unlock nearly every calculus problem.
AP Calculus BC
Why These Four Functions Matter
Nearly every function you encounter in AP Calculus can be built from trigonometric and exponential/logarithmic pieces. Knowing their derivatives by heart lets you apply the chain rule, product rule, and quotient rule to solve complex problems quickly and confidently.
Quick Refresher: The Derivative
The derivative of \(f(x)\) at a point measures the instantaneous rate of change: the slope of the tangent line. Formally:
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
Visualizing \(\sin x\) and Its Derivative
Core Rules: Trigonometric Derivatives
Sine
$$\frac{d}{dx}[\sin x] = \cos x$$
Cosine
$$\frac{d}{dx}[\cos x] = -\sin x$$
Visualizing \(e^x\) and \(\ln x\)
Core Rules: Exponential & Logarithmic
Exponential
$$\frac{d}{dx}[e^x] = e^x$$
Natural Log
$$\frac{d}{dx}[\ln x] = \frac{1}{x}$$
Key Distinction: Signs & Patterns
| Function | Derivative | Sign Change? | |
|---|---|---|---|
| \(\sin x\) | oscillates | \(\cos x\) | No |
| \(\cos x\) | oscillates | \(-\sin x\) | Yes (negative) |
| \(e^x\) | always positive | \(e^x\) | No |
| \(\ln x\) | increasing | \(\frac{1}{x}\) | No (for \(x > 0\)) |
Why \(\frac{d}{dx}[\sin x] = \) \(\cos x\)?
1
Start from the limit definition. Write \(\displaystyle\lim_{h\to 0}\frac{\sin(x+h)-\sin x}{h}\).
2
Apply the angle-addition formula: \(\sin(x+h)=\sin x\cos h+\cos x\sin h\).
3
Simplify. You get \(\sin x\cdot\frac{\cos h - 1}{h}+\cos x\cdot\frac{\sin h}{h}\).
4
Use special limits: \(\displaystyle\lim_{h\to 0}\frac{\sin h}{h}=1\) and \(\displaystyle\lim_{h\to 0}\frac{\cos h -1}{h}=0\). Result: \(\cos x\).
How to Apply These Rules
1
Identify the outer function. Is it \(\sin(\cdot)\), \(\cos(\cdot)\), \(e^{(\cdot)}\), or \(\ln(\cdot)\)?
2
Write down its derivative formula from the four core rules.
3
Check for a chain rule. If the argument is not just \(x\), multiply by the derivative of the inner function.
4
Simplify. Combine like terms, factor where possible, and verify the sign.
Example 1: Basic Trig Derivative
Find \(\displaystyle\frac{d}{dx}\bigl[5\sin x - 2\cos x\bigr]\).
1
Differentiate term by term. Use linearity: \(\frac{d}{dx}[af(x)] = a \cdot f'(x)\).
2
First term: \(\frac{d}{dx}[5\sin x] = 5\cos x\).
3
Second term: \(\frac{d}{dx}[-2\cos x] = -2(-\sin x) = 2\sin x\).
Example 2: \(e^x\) and \(\ln x\) Together
Find \(\displaystyle\frac{d}{dx}\bigl[e^x + 3\ln x\bigr]\).
1
First term: \(\frac{d}{dx}[e^x] = e^x\). It's its own derivative.
2
Second term: \(\frac{d}{dx}[3\ln x] = 3\cdot\frac{1}{x} = \frac{3}{x}\).
Example 3: Chain Rule in Action
Find \(\displaystyle\frac{d}{dx}\bigl[e^{3x}\sin x\bigr]\).
1
Product rule: \((uv)' = u'v + uv'\). Let \(u = e^{3x}\) and \(v = \sin x\).
2
Chain rule for \(u\): \(\frac{d}{dx}[e^{3x}] = 3e^{3x}\). Inner function \(3x\) has derivative \(3\).
3
Assemble: \(3e^{3x}\sin x + e^{3x}\cos x\).
Example 4: Composite \(\ln\) Function
Find \(\displaystyle\frac{d}{dx}\bigl[\ln(\cos x)\bigr]\).
1
Outer function: \(\ln(\cdot)\), its derivative is \(\frac{1}{(\cdot)}\).
2
Inner function: \(\cos x\), its derivative is \(-\sin x\).
3
Chain rule: \(\frac{1}{\cos x}\cdot(-\sin x) = -\frac{\sin x}{\cos x}\).
Common Mistakes to Avoid
✕ Forgetting the Negative
Writing \(\frac{d}{dx}[\cos x] = \sin x\) instead of \(-\sin x\). The negative is critical.
✕ Dropping the Chain Rule
Writing \(\frac{d}{dx}[e^{3x}] = e^{3x}\) without the factor of \(3\). The inner derivative must be included.
✕ Wrong \(\ln\) Derivative
Writing \(\frac{d}{dx}[\ln x] = x\) instead of \(\frac{1}{x}\). The derivative shrinks, not grows.
Complete Reference Table
| Function \(f(x)\) | Derivative \(f'(x)\) | Domain |
|---|---|---|
| \(\sin x\) | \(\cos x\) | All \(\mathbb{R}\) |
| \(\cos x\) | \(-\sin x\) | All \(\mathbb{R}\) |
| \(e^x\) | \(e^x\) | All \(\mathbb{R}\) |
| \(\ln x\) | \(\dfrac{1}{x}\) | \(x > 0\) |
Question 1 of 5
What is \(\displaystyle\frac{d}{dx}[\cos x]\)?
The derivative of \(\cos x\) is \(-\sin x\). Remember: "cosine goes negative." This is one of the most frequently tested facts on the AP exam.
Question 2 of 5
What is \(\displaystyle\frac{d}{dx}[e^{5x}]\)?
By the chain rule, \(\frac{d}{dx}[e^{5x}] = e^{5x} \cdot 5 = 5e^{5x}\). The derivative of the inner function \(5x\) is \(5\), which multiplies the exponential.
Question 3 of 5
What is \(\displaystyle\frac{d}{dx}[\ln(x^2)]\)?
Using the chain rule: \(\frac{d}{dx}[\ln(x^2)] = \frac{1}{x^2}\cdot 2x = \frac{2}{x}\). Alternatively, simplify first: \(\ln(x^2) = 2\ln x\), so the derivative is \(2 \cdot \frac{1}{x} = \frac{2}{x}\).
Question 4 of 5
What is \(\displaystyle\frac{d}{dx}[\sin x \cdot \ln x]\)?
Apply the product rule: \((uv)' = u'v + uv'\). With \(u=\sin x\) and \(v=\ln x\): \(\cos x \cdot \ln x + \sin x \cdot \frac{1}{x}\). This gives \(\cos x\,\ln x + \frac{\sin x}{x}\).
Question 5 of 5
If \(f(x) = e^x - \cos x\), what is \(f'(0)\)?
First, find the derivative: \(f'(x) = \frac{d}{dx}[e^x - \cos x] = e^x - (-\sin x) = e^x + \sin x\). Then, evaluate at \(x=0\): \(f'(0) = e^0 + \sin 0 = 1 + 0 = 1\).