The Power Rule for Derivatives

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1. 1. The rule0:00
   $$\frac{d}{dx}\, x^n = n\,x^{n-1}$$

   bring the power down, then lower it by one

2. 2. Term by term0:11
   1. $$\frac{d}{dx}\left(3x^4 - 2x^2 + 7x\right)$$
   2. $$= 3 \cdot 4x^3 - 2 \cdot 2x + 7$$
   3. $$= 12x^3 - 4x + 7$$
3. 3. Slope at a point0:31
   1. $$f'(1) = 12(1)^3 - 4(1) + 7$$
   2. $$f'(1) = 15$$
4. 4. Recap0:40
   • ✓$\frac{d}{dx} x^n = n x^{n-1}$
   • ✓Constants multiply through; $\frac{d}{dx}(7x) = 7$.
   • ✓$f'(a)$ is the tangent slope at $x = a$.

Power, product, quotient and chain rules.

• Derivative Calculator: try the same kind of problem with your own numbers.
• Implicit Differentiation Calculator: try the same kind of problem with your own numbers.
• More calculus practice problems
• Limits Study Guide (study guide)
• AP Calculus AB Unit 2: Differentiation: Definition and Basic Rules: the AP unit this lesson supports.
• Calculus calculators and topics
• Step-by-step solver: check your own answer with hints.

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