Now that you understand what a derivative is, it's time to learn how to compute derivatives quickly without using the limit definition every time. These basic rules are the foundation for all differentiation you'll do in AP Calculus.
🎯 In this section you will learn
- The Power Rule: $\frac{d}{dx}x^n = nx^{n-1}$
- The Constant Multiple Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$
- The Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
- Derivatives of $e^x$ and $\ln x$
📋 The Basic Rules (Reference Table)
| Rule | Formula | Example |
|---|
| Power Rule | $\frac{d}{dx}x^n = nx^{n-1}$ | $\frac{d}{dx}x^5 = 5x^4$ |
| Constant Multiple Rule | $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$ | $\frac{d}{dx}(3x^2) = 6x$ |
| Sum Rule | $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$ | $\frac{d}{dx}(x^2 + x^3) = 2x + 3x^2$ |
| Difference Rule | $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$ | $\frac{d}{dx}(x^2 - x^3) = 2x - 3x^2$ |
| Derivative of $e^x$ | $\frac{d}{dx}e^x = e^x$ | $\frac{d}{dx}(2e^x) = 2e^x$ |
| Derivative of $\ln x$ | $\frac{d}{dx}\ln x = \frac{1}{x}$ | $\frac{d}{dx}(5\ln x) = \frac{5}{x}$ |
📌 The Power Rule in Depth
The power rule is the workhorse of differentiation. For any real number $n$:
$$ \frac{d}{dx}x^n = n \cdot x^{n-1} $$
This works for positive integers, negative integers, fractions — any real exponent!
- $\frac{d}{dx}x^4 = 4x^3$
- $\frac{d}{dx}x^7 = 7x^6$
- $\frac{d}{dx}x^{10} = 10x^9$
- $\frac{d}{dx}x^{-3} = -3x^{-4} = -\frac{3}{x^4}$
- $\frac{d}{dx}x^{-1} = -1 \cdot x^{-2} = -\frac{1}{x^2}$
- $\frac{d}{dx}x^{-2} = -2x^{-3} = -\frac{2}{x^3}$
- $\frac{d}{dx}x^{1/2} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$
- $\frac{d}{dx}x^{2/3} = \frac{2}{3}x^{-1/3} = \frac{2}{3\sqrt[3]{x}}$
- $\frac{d}{dx}\sqrt{x} = \frac{d}{dx}x^{1/2} = \frac{1}{2\sqrt{x}}$
📌 Constant Multiple Rule
Constants "come along for the ride" — they multiply the derivative:
$$ \frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}f(x) $$
- $\frac{d}{dx}(5x^2) = 5 \cdot 2x = 10x$
- $\frac{d}{dx}(-3x^4) = -3 \cdot 4x^3 = -12x^3$
- $\frac{d}{dx}(7e^x) = 7e^x$
- $\frac{d}{dx}(4\ln x) = 4 \cdot \frac{1}{x} = \frac{4}{x}$
📌 Sum and Difference Rules
You can differentiate term by term:
$$ \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) $$
$$ \frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x) $$
Find $\frac{d}{dx}(3x^5 - 2x^3 + 4x - 7)$.
①
Differentiate each term
$\frac{d}{dx}3x^5 = 15x^4$
②
Next term
$\frac{d}{dx}(-2x^3) = -6x^2$
③
Next term
$\frac{d}{dx}(4x) = 4$
④
Constant term
$\frac{d}{dx}(-7) = 0$
⑤
Combine
$\frac{d}{dx}(3x^5 - 2x^3 + 4x - 7) = 15x^4 - 6x^2 + 4$
📌 Derivatives of $e^x$ and $\ln x$
These two functions have special derivatives that you must memorize:
$$ \frac{d}{dx}e^x = e^x \qquad \text{and} \qquad \frac{d}{dx}\ln x = \frac{1}{x} \;\; (x > 0) $$
- $\frac{d}{dx}e^x = e^x$
- $\frac{d}{dx}(5e^x) = 5e^x$
- $\frac{d}{dx}(e^x + x^2) = e^x + 2x$
- $\frac{d}{dx}\ln x = \frac{1}{x}$
- $\frac{d}{dx}(3\ln x) = \frac{3}{x}$
- $\frac{d}{dx}(x^2 + \ln x) = 2x + \frac{1}{x}$
⚠️ Common Mistakes
- Forgetting the power rule: $\frac{d}{dx}x^n = n \cdot x^{n-1}$, not $n \cdot x^{n}$.
- Constants disappear: Derivative of a constant is $0$, not $1$.
- $e^x$ is special: Its derivative is itself — many students mistakenly think $\frac{d}{dx}e^x = e^{x-1}$ or $x e^{x-1}$.
- $\ln x$ domain: Only defined for $x > 0$.
🔍 Key Takeaways
- Power Rule: $\frac{d}{dx}x^n = nx^{n-1}$ (works for any real $n$).
- Constants multiply the derivative: $\frac{d}{dx}[c f(x)] = c f'(x)$.
- Derivative of a constant is $0$.
- You can differentiate term by term: $(f \pm g)' = f' \pm g'$.
- Memorize: $\frac{d}{dx}e^x = e^x$ and $\frac{d}{dx}\ln x = \frac{1}{x}$.
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