Differentiating polynomials
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Exercise:
Differentiate the following polynomials with respect to x. nprvmulticols abclist abc fx x abc fx x^ abc fx x^ abc fx x^ abc fx x^ + x^ abc fx x^ - x^ abc fx -x^ + x abc fx x^ - x^ + x^ abc fx x^ + x^ + x + abc fx x^ - x^ + x^ - x^ + x abc fx x^ + x^ - x^ + abc fx xx^ + abc fx x^ + xx - abc fx x^x^ - x + abc fx x^ - xx^ + abc fx x^x - x + abc fx x^ + ^ abc fx x^ - x + ^ abc fx x^ + x^ + x + ^ abc fx x^ - x^x + abc fx xx - x + x^ + abc fx x^ - x^x^ + abc fx x^ + ^x^ - x + abc fx x^ + x + ^x^ + x + ^ abclist nprvmulticols
Solution:
abclist abc Differentiate fx x . textbfSolution: f'x abc Differentiate fx x^ . textbfSolution: f'x x abc Differentiate fx x^ . textbfSolution: f'x x^ abc Differentiate fx x^ . textbfSolution: f'x x^ abc Differentiate fx x^ + x^ . textbfSolution: f'x x^ + x^ abc Differentiate fx x^ - x^ . textbfSolution: f'x x^ - x abc Differentiate fx -x^ + x . textbfSolution: f'x -x^ + abc Differentiate fx x^ - x^ + x^ . textbfSolution: f'x x^ - x^ + x abc Differentiate fx x^ + x^ + x + . textbfSolution: f'x x^ + x + abc Differentiate fx x^ - x^ + x^ - x^ + x . textbfSolution: f'x x^ - x^ + x^ - x + abc Differentiate fx x^ + x^ - x^ + . textbfSolution: f'x x^ + x^ - x^ abc Differentiate fx xx^ + . textbfSolution: f'x x x + x^ + x^ + x^ + x^ + abc Differentiate fx x^ + xx - . textbfSolution: f'x x + x - + x^ + x x + x - + x^ + x abc Differentiate fx x^x^ - x + . textbfSolution: f'x xx^ - x + + x^x - abc Differentiate fx x^ - xx^ + . textbfSolution: f'x x^ - x^ + + x^ - xx abc Differentiate fx x^x - x + . textbfSolution: Use product rule on three functions: Let u x^ v x - w x + then f'x u'vw + uv'w + uvw' f'x x^x - x + + x^x + + x^x - abc Differentiate fx x^ + ^ . textbfSolution: Use chain rule: f'x x^ + x xx^ + abc Differentiate fx x^ - x + ^ . textbfSolution: Chain rule: f'x x^ - x + ^ x - abc Differentiate fx x^ + x^ + x + ^ . textbfSolution: Chain rule: f'x x^ + x^ + x + x^ + x + abc Differentiate fx x^ - x^x + . textbfSolution: Product rule: Let u x^ - x^ v x + then u' x^ - xx - so f'x u'v + uv' x^ - xx - x + + x^ - x^ abc Differentiate fx xx - x + x^ + . textbfSolution: Product rule with multiple functions -- expand or apply generalized product rule best to expand first if needed in practice. Partial answer format: Differentiate one term at a time holding others constant. abc Differentiate fx x^ - x^x^ + . textbfSolution: Let u x^ - x^ v x^ + Then u' x^ - xx^ - so f'x u'v + uv' x^ - xx^ - x^ + + x^ - x^x abc Differentiate fx x^ + ^x^ - x + . textbfSolution: Product rule: Let u x^ + ^ v x^ - x + Then u' x^ + x v' x^ - So f'x u'v + uv' xx^ + x^ - x + + x^ + ^x^ - abc Differentiate fx x^ + x + ^x^ + x + ^ . textbfSolution: Product + chain rule: Let u x^ + x + ^ v x^ + x + ^ Then u' x^ + x + ^x + v' x^ + x + x^ + So: f'x u'v + uv' abclist
Differentiate the following polynomials with respect to x. nprvmulticols abclist abc fx x abc fx x^ abc fx x^ abc fx x^ abc fx x^ + x^ abc fx x^ - x^ abc fx -x^ + x abc fx x^ - x^ + x^ abc fx x^ + x^ + x + abc fx x^ - x^ + x^ - x^ + x abc fx x^ + x^ - x^ + abc fx xx^ + abc fx x^ + xx - abc fx x^x^ - x + abc fx x^ - xx^ + abc fx x^x - x + abc fx x^ + ^ abc fx x^ - x + ^ abc fx x^ + x^ + x + ^ abc fx x^ - x^x + abc fx xx - x + x^ + abc fx x^ - x^x^ + abc fx x^ + ^x^ - x + abc fx x^ + x + ^x^ + x + ^ abclist nprvmulticols
Solution:
abclist abc Differentiate fx x . textbfSolution: f'x abc Differentiate fx x^ . textbfSolution: f'x x abc Differentiate fx x^ . textbfSolution: f'x x^ abc Differentiate fx x^ . textbfSolution: f'x x^ abc Differentiate fx x^ + x^ . textbfSolution: f'x x^ + x^ abc Differentiate fx x^ - x^ . textbfSolution: f'x x^ - x abc Differentiate fx -x^ + x . textbfSolution: f'x -x^ + abc Differentiate fx x^ - x^ + x^ . textbfSolution: f'x x^ - x^ + x abc Differentiate fx x^ + x^ + x + . textbfSolution: f'x x^ + x + abc Differentiate fx x^ - x^ + x^ - x^ + x . textbfSolution: f'x x^ - x^ + x^ - x + abc Differentiate fx x^ + x^ - x^ + . textbfSolution: f'x x^ + x^ - x^ abc Differentiate fx xx^ + . textbfSolution: f'x x x + x^ + x^ + x^ + x^ + abc Differentiate fx x^ + xx - . textbfSolution: f'x x + x - + x^ + x x + x - + x^ + x abc Differentiate fx x^x^ - x + . textbfSolution: f'x xx^ - x + + x^x - abc Differentiate fx x^ - xx^ + . textbfSolution: f'x x^ - x^ + + x^ - xx abc Differentiate fx x^x - x + . textbfSolution: Use product rule on three functions: Let u x^ v x - w x + then f'x u'vw + uv'w + uvw' f'x x^x - x + + x^x + + x^x - abc Differentiate fx x^ + ^ . textbfSolution: Use chain rule: f'x x^ + x xx^ + abc Differentiate fx x^ - x + ^ . textbfSolution: Chain rule: f'x x^ - x + ^ x - abc Differentiate fx x^ + x^ + x + ^ . textbfSolution: Chain rule: f'x x^ + x^ + x + x^ + x + abc Differentiate fx x^ - x^x + . textbfSolution: Product rule: Let u x^ - x^ v x + then u' x^ - xx - so f'x u'v + uv' x^ - xx - x + + x^ - x^ abc Differentiate fx xx - x + x^ + . textbfSolution: Product rule with multiple functions -- expand or apply generalized product rule best to expand first if needed in practice. Partial answer format: Differentiate one term at a time holding others constant. abc Differentiate fx x^ - x^x^ + . textbfSolution: Let u x^ - x^ v x^ + Then u' x^ - xx^ - so f'x u'v + uv' x^ - xx^ - x^ + + x^ - x^x abc Differentiate fx x^ + ^x^ - x + . textbfSolution: Product rule: Let u x^ + ^ v x^ - x + Then u' x^ + x v' x^ - So f'x u'v + uv' xx^ + x^ - x + + x^ + ^x^ - abc Differentiate fx x^ + x + ^x^ + x + ^ . textbfSolution: Product + chain rule: Let u x^ + x + ^ v x^ + x + ^ Then u' x^ + x + ^x + v' x^ + x + x^ + So: f'x u'v + uv' abclist