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Derivative by definition worksheets

Differentiate polynomial, rational, signed-power composite and trigonometric functions from the limit definition.

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Use the derivative definition, cancel or apply a standard-limit identity, then take h to zero.

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Practice with exact answers

Keep fractions, radicals and pi exact. Round only when a question asks for a decimal answer, and check every original domain restriction.

Learning objective

What this activity builds

Use a true h-to-zero difference quotient with an algebraic or trigonometric cancellation witness.

Worked example

How does a rate arise from a function?

Compare function values, then divide by the matching input change when the task asks for a rate.

Delta y, average rate, a derivative limit and an instantaneous rate are related but distinct quantities. Read the requested form before calculating.

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  • Keep exact values through the final step.
  • Retain every original domain restriction.
  • Treat trigonometric inputs as radians.
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Math SheetDerivative from the DefinitionUse the limit definition and show cancellation or a standard-limit identity.1.y = (x^2+1)/(x-2)2.y = 1(3x^2+1)^53.y = (x^2-6x)/(2x^2-12x+18)4.y = (1/2)/(x^2-4x+4)+(-3)/(x^2)5.y = 4sin(-6x+3)math-sheet.comPage 1 of 2
Math SheetDerivative from the DefinitionUse the limit definition and show cancellation or a standard-limit identity.6.y = 2x^3-x+47.y = -2cos(3x-12)8.y = x^4-2x^2+39.y = -x^2-3x-510.y = 1(-x^3-x)^-3math-sheet.comPage 2 of 2

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