The power rule and constant rule are key tools for finding derivatives of polynomials. These rules simplify the process of differentiation by providing straightforward steps for each term in a polynomial function.

By applying these rules, you can quickly determine how a polynomial function changes. The power rule reduces the degree of each term, while the constant rule eliminates constant terms, resulting in a new polynomial that represents the rate of change of the original function.

Power Rule and Constant Rule

Power rule for polynomial derivatives

• States for a function $f(x) = x^n$, its derivative is $f'(x) = nx^{n-1}$
  • $f(x) = x^4$, then $f'(x) = 4x^3$
• To apply to a term in a polynomial, multiply coefficient by exponent and decrease exponent by 1
  • $f(x) = 3x^5$, then $f'(x) = 3 \cdot 5x^{5-1} = 15x^4$
• When polynomial has multiple terms, apply power rule to each term separately
  • $f(x) = 2x^3 + 4x^2 - 5x$, then $f'(x) = 2 \cdot 3x^{3-1} + 4 \cdot 2x^{2-1} - 5 \cdot 1x^{1-1} = 6x^2 + 8x - 5$

Constant rule in differentiation

• States derivative of a constant function is always 0
  • $f(x) = 7$, then $f'(x) = 0$
• Applies to any constant term in a polynomial function
  • $f(x) = 3x^2 + 5$, then derivative of constant term 5 is 0
• Constant terms remain unchanged in the derivative ($2x^3 + 4$ becomes $6x^2 + 0$)

Combining rules for polynomials

• When differentiating a polynomial, apply power rule to each term with a variable and constant rule to constant terms
  • $f(x) = 4x^3 - 2x + 6$, then $f'(x) = 4 \cdot 3x^{3-1} - 2 \cdot 1x^{1-1} + 0 = 12x^2 - 2$
• Add derivatives of each term together to find final derivative of polynomial
  • $(3x^4 - x^2 + 2x - 5)' = 12x^3 - 2x + 2 - 0$

Polynomial degree vs derivative

• Degree of polynomial is highest exponent of variable
  • $f(x) = 3x^4 + 2x^2 - 5x + 1$, degree is 4
• When differentiating, degree of resulting derivative is one less than original polynomial
  • $f(x) = 3x^4 + 2x^2 - 5x + 1$, then $f'(x) = 12x^3 + 4x - 5$, which has degree 3
• Relationship holds for all polynomials, as power rule decreases exponent of each term by 1
  • $(7x^5 - 4x^3 + 6x)' = 35x^4 - 12x^2 + 6$, degree reduced from 5 to 4