5.2 Sum and difference rules

The sum and difference rules for derivatives simplify complex calculations by breaking functions into manageable parts. These rules state that the derivative of a sum or difference equals the sum or difference of individual derivatives.

Applying these rules allows us to tackle intricate functions step-by-step. We can differentiate each term separately, then combine the results. This approach is especially useful when dealing with polynomials, trigonometric functions, and other complex expressions.

Sum and Difference Rules

Sum rule for derivatives

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• States the derivative of a sum of functions equals the sum of the derivatives of each function
  • For differentiable functions $f(x)$ and $g(x)$, the derivative of their sum is: $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$
  • Extends to the sum of any number of differentiable functions ($h(x)$, $k(x)$, etc.)
• Differentiate each function separately and add the results together to find the derivative of a sum of functions
• Find the derivative of $f(x) = x^3 + \sin(x)$
  • $\frac{d}{dx}f(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(\sin(x))$ (differentiate each term)
  • $\frac{d}{dx}f(x) = 3x^2 + \cos(x)$ (apply power rule and trigonometric differentiation)

Difference rule for derivatives

• States the derivative of a difference of functions equals the difference of the derivatives of each function
  • For differentiable functions $f(x)$ and $g(x)$, the derivative of their difference is: $\frac{d}{dx}(f(x) - g(x)) = \frac{d}{dx}f(x) - \frac{d}{dx}g(x)$
• Differentiate each function separately and subtract the results to find the derivative of a difference of functions
• Find the derivative of $f(x) = e^x - \ln(x)$
  • $\frac{d}{dx}f(x) = \frac{d}{dx}(e^x) - \frac{d}{dx}(\ln(x))$ (differentiate each term)
  • $\frac{d}{dx}f(x) = e^x - \frac{1}{x}$ (apply exponential and logarithmic differentiation rules)

Combining differentiation rules

• Sum and difference rules can be used with other differentiation rules (constant multiple rule, product rule, chain rule) for more complex functions
• Break down complex functions into simpler components and apply appropriate differentiation rules to each component
• Find the derivative of $f(x) = (3x^2 + 2x)(x^3 - 4)$
  1. Use the product rule, $\frac{d}{dx}(u(x)v(x)) = u(x)\frac{d}{dx}v(x) + v(x)\frac{d}{dx}u(x)$, where $u(x) = 3x^2 + 2x$ and $v(x) = x^3 - 4$
  2. $\frac{d}{dx}u(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(2x) = 6x + 2$ (sum rule and constant multiple rule)
  3. $\frac{d}{dx}v(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(4) = 3x^2 - 0$ (difference rule and constant rule)
  4. $\frac{d}{dx}f(x) = (3x^2 + 2x)(3x^2) + (x^3 - 4)(6x + 2)$ (combine results using product rule)

Applications of sum and difference rules

• Particularly useful for functions composed of multiple terms, each differentiable separately
• Allow breaking down complex functions into simpler components, differentiating each independently, and combining results using addition or subtraction
• Identifying functions as sums or differences of other functions simplifies the differentiation process
• Find the derivative of $f(x) = (2x + 1)^3 - \sqrt{x} + \sin(x)$
  • Recognize the function as a difference of three terms: $(2x + 1)^3$, $\sqrt{x}$, and $\sin(x)$
  • Apply chain rule to first term, power rule to second term, and trigonometric differentiation to third term
  • Combine results using difference rule for final derivative: $[f'(x)](https://www.fiveableKeyTerm:f'(x)) = 6(2x+1)^2 - \frac{1}{2\sqrt{x}} + \cos(x)$