5.1 Power rule and constant rule

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