Derivative Rules: Constant, Sum, Difference, and Constant Multiple AB & BC
Differentiation is linear: constants have derivative 0, a constant multiplier passes through unchanged, and a sum or difference is differentiated one term at a time with the signs preserved. So $\frac{d}{dx}[3x^5 - 2x^2 + 7x - 9] = 15x^4 - 4x + 7$. Expressions that are not already sums of powers are rewritten first, splitting a single-term denominator or treating a constant denominator as a constant multiple.
The slips are arithmetic and scope. A coefficient gets dropped or multiplied wrongly, a trailing constant survives into the answer, or only the leading term is differentiated while the rest are copied down. And linearity gets stretched onto operations it does not cover: a product or quotient handled term by term, or a term such as $2^x$ pushed through the power rule because the sum rule delivered it.
The work
3 ways in · any order
Lesson
Derivative Rules: Constant, Sum, Difference, and Constant Multiple
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Runs the constant, constant multiple, sum and difference rules across polynomials, rewrites single-term denominators and constant denominators into power form, and marks the boundary where linearity stops and the product and quotient rules begin.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: dropping or mis-multiplying a coefficient, leaving a constant term alive in the answer, or differentiating only part of a sum, and stretching the power rule onto a term the sum rule merely handed you.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.