3.4 Basic Differentiation Rules

Derivatives are powerful tools for understanding rates of change. In this part, we'll learn the basic rules for finding derivatives, including the constant rule, power rule, and sum rule. These form the foundation for more complex differentiation techniques.

We'll also dive into advanced rules like the product rule, quotient rule, and chain rule. These allow us to tackle more complicated functions and compositions. Understanding these rules is crucial for mastering differentiation and applying it to real-world problems.

Basic Differentiation Rules

Constant and Power Rules

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• Constant rule states the derivative of a constant is always 0 ($\frac{d}{dx}(c) = 0$)
• Power rule is used to find the derivative of a variable raised to a power ($\frac{d}{dx}(x^n) = nx^{n-1}$)
  • Multiplies the exponent by the coefficient
  • Subtracts 1 from the exponent
  • Example: $\frac{d}{dx}(x^3) = 3x^2$
• Constant multiple rule allows constants to be factored out of the derivative ($\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))$)
  • Example: $\frac{d}{dx}(5x^2) = 5\frac{d}{dx}(x^2) = 5(2x) = 10x$

Sum and Difference Rules

• Sum rule states the derivative of a sum is the sum of the derivatives ($\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$)
  • Allows each term to be differentiated separately and then added together
  • Example: $\frac{d}{dx}(x^2 + 3x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(3x) = 2x + 3$
• Difference rule states the derivative of a difference is the difference of the derivatives ($\frac{d}{dx}(f(x) - g(x)) = \frac{d}{dx}(f(x)) - \frac{d}{dx}(g(x))$)
  • Allows each term to be differentiated separately and then subtracted
  • Example: $\frac{d}{dx}(x^3 - 2x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(2x) = 3x^2 - 2$

Advanced Differentiation Rules

Product and Quotient Rules

• Product rule is used to find the derivative of the product of two functions ($\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}(g(x)) + g(x)\frac{d}{dx}(f(x))$)
  • Multiplies the first function by the derivative of the second
  • Adds that to the product of the second function and the derivative of the first
  • Example: $\frac{d}{dx}(x^2\sin(x)) = x^2\frac{d}{dx}(\sin(x)) + \sin(x)\frac{d}{dx}(x^2) = x^2\cos(x) + 2x\sin(x)$
• Quotient rule is used to find the derivative of the quotient of two functions ($\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{g(x)\frac{d}{dx}(f(x)) - f(x)\frac{d}{dx}(g(x))}{[g(x)]^2}$)
  • Multiplies the denominator by the derivative of the numerator
  • Subtracts the numerator times the derivative of the denominator
  • Divides that difference by the square of the denominator
  • Example: $\frac{d}{dx}(\frac{x^2}{x+1}) = \frac{(x+1)\frac{d}{dx}(x^2) - x^2\frac{d}{dx}(x+1)}{(x+1)^2} = \frac{2x(x+1)-x^2}{(x+1)^2}$

Chain Rule

• Chain rule is used to find the derivative of a composition of functions ($\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$)
  • Multiplies the derivative of the outer function evaluated at the inner function
  • Multiplies that by the derivative of the inner function
  • Requires you to identify the "inner" and "outer" functions
  • Example: $\frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x$
    • $\sin(x)$ is the outer function, $x^2$ is the inner function

Higher-Order Derivatives

Finding Higher-Order Derivatives

• Higher-order derivatives are derivatives of derivatives
  • The second derivative is the derivative of the first derivative ($f''(x) = \frac{d}{dx}([f'(x)](https://www.fiveableKeyTerm:f'(x)))$)
  • The third derivative is the derivative of the second derivative ($f'''(x) = \frac{d}{dx}(f''(x))$)
  • Notation: $f^{(n)}(x)$ represents the nth derivative of $f(x)$
• Higher-order derivatives are found by repeatedly applying differentiation rules
  • Example: If $f(x) = x^4$, then:
    • $f'(x) = 4x^3$
    • $f''(x) = 12x^2$
    • $f'''(x) = 24x$
    • $f^{(4)}(x) = 24$
• Physical interpretations often involve rates of change
  • First derivative of position is velocity
  • Second derivative of position (first derivative of velocity) is acceleration