The and are key tools for finding derivatives of polynomials. These rules simplify the process of differentiation by providing straightforward steps for each term in a .
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
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States for a function f(x)=xn, its derivative is [f′(x)](https://www.fiveableKeyTerm:f′(x))=nxn−1
f(x)=x4, then f′(x)=4x3
To apply to a term in a polynomial, multiply coefficient by exponent and decrease exponent by 1
f(x)=3x5, then f′(x)=3⋅5x5−1=15x4
When polynomial has multiple terms, apply power rule to each term separately
f(x)=2x3+4x2−5x, then f′(x)=2⋅3x3−1+4⋅2x2−1−5⋅1x1−1=6x2+8x−5
Constant rule in differentiation
States function is always 0
f(x)=7, then f′(x)=0
Applies to any constant term in a polynomial function
f(x)=3x2+5, then derivative of constant term 5 is 0
Constant terms remain unchanged in the derivative (2x3+4 becomes 6x2+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)=4x3−2x+6, then f′(x)=4⋅3x3−1−2⋅1x1−1+0=12x2−2
Add derivatives of each term together to find final derivative of polynomial
(3x4−x2+2x−5)′=12x3−2x+2−0
Polynomial degree vs derivative
is highest exponent of variable
f(x)=3x4+2x2−5x+1, degree is 4
When differentiating, is one less than original polynomial
f(x)=3x4+2x2−5x+1, then f′(x)=12x3+4x−5, which has degree 3
Relationship holds for all polynomials, as power rule decreases exponent of each term by 1
(7x5−4x3+6x)′=35x4−12x2+6, degree reduced from 5 to 4