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2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Syllabus
2020
Topic
2.6
Level
—

Learning objectives

—FUN-3.A—Calculate derivatives of familiar functions—Topic 2.6• FUN-3.A Calculate derivatives of familiar functions.• FUN-3.A.2 Sums, differences, and constant multiples of functions can be differentiated using derivative rules.• FUN-3.A.3 The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

Differentiate Polynomials Term by Term

Derivatives distribute across addition and subtraction, and constant factors stay attached. This lets you differentiate a polynomial one term at a time instead of returning to the limit definition.

\frac{d}{dx}(c)=0,\qquad (f+g)'=f'+g',\qquad (f-g)'=f'-g',\qquad (cf)'=cf'

For each nonconstant term axnax^n, keep the coefficient aa and apply the power rule: axn↦anxn−1ax^n\mapsto an x^{n-1}. Preserve the operation sign between terms, and replace every constant term by 00.

If p(x)=4x5−3x2+7p(x)=4x^5-3x^2+7, then p′(x)=4(5x4)−3(2x)+0=20x4−6x.p'(x)=4(5x^4)-3(2x)+0=20x^4-6x. The coefficient of each power is multiplied by its old exponent, the exponent decreases by one, and the constant disappears.

The constant-multiple rule applies when the multiplier is constant with respect to xx. Do not treat a product of two variable functions as a constant multiple, and do not lose a negative sign when differentiating a difference.