Differentiating polynomials

Japanese school year: Math II

What you learn

This topic focuses on computing derivatives of polynomial functions using power and linearity differentiation rules. Mastered techniques serve as the fundamental groundwork for determining instantaneous rates of change and analyzing graph slopes. A solid background in monomial operations, polynomial algebra, and exponent rules is necessary before starting.

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Key points

This rule is used to differentiate a power term $x^n$ . Bring the exponent $n$ to the front as a multiplier and decrease the power by 1 to get $n x^{n-1}$ (for example, $x^3$ differentiates to $3x^2$ ).

(x^n)' = n x^{n-1}

This formula shows that the derivative of any constant $c$ is 0. Since a constant is a fixed number that never changes, its rate of change is always 0.

(c)' = 0

This formula is used when a function $f(x)$ is multiplied by a constant $k$ . Keep the constant $k$ as it is and differentiate only the function part $f(x)$ .

\{k f(x)\}' = k f'(x)

This formula is used to differentiate the sum of two functions, $f(x)$ and $g(x)$ . You can differentiate $f(x)$ and $g(x)$ separately, then add their derivatives together.

\{f(x) + g(x)\}' = f'(x) + g'(x)

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Practice 1Coming soon