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Question Number 157836 by zakirullah last updated on 28/Oct/21

find the indicated higher order derivative of the following function f(x) = (x^3 +4x−5)^4 , f(x)^(iv)

$${find}\:{the}\:{indicated}\:{higher}\:{order}\:{derivative} \\ $$$${of}\:{the}\:{following}\:{function} \\ $$$${f}\left({x}\right)\:=\:\left({x}^{\mathrm{3}} +\mathrm{4}{x}−\mathrm{5}\right)^{\mathrm{4}} ,\:{f}\left({x}\right)^{{iv}} \\ $$

Answered by tounghoungko last updated on 29/Oct/21

f(x)=(x−1)^4 (x^2 +x+5)^4 (d^4 y/dx^4 ) = (d^4 /dx^4 )((x−1)^4 )(x^2 +x+5)^4 +4(d^3 /dx^3 )((x−1))^4 .(d/dx)(x^2 +x+5)^4 + 6 (d^2 /dx^2 )((x−1))^4 (d^2 /dx^2 )(x^2 +x+5)^4 +4(d/dx)((x−1))^4 (d^3 /dx^3 )(x^2 +x+5)^4 + 4(x−1)^4 (d^4 /dx^4 )(x^2 +x+5)^4 = 24(x^2 +x+5)^4 +96(x−1)(d/dx)(x^2 +x+5)^4 +72(x−1)^2 (d^2 /dx^2 )(x^2 +x+5)^4 + 16(x−1)^3 (d^3 /dx^3 )(x^2 +x+5)^4 +4(x−1)^4 (d^4 /dx^4 )(x^2 +x+5)^4

$${f}\left({x}\right)=\left({x}−\mathrm{1}\right)^{\mathrm{4}} \left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} \\ $$$$\frac{{d}^{\mathrm{4}} {y}}{{dx}^{\mathrm{4}} }\:=\:\frac{{d}^{\mathrm{4}} }{{dx}^{\mathrm{4}} }\left(\left({x}−\mathrm{1}\right)^{\mathrm{4}} \right)\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} +\mathrm{4}\frac{{d}^{\mathrm{3}} }{{dx}^{\mathrm{3}} }\left(\left({x}−\mathrm{1}\right)\right)^{\mathrm{4}} .\frac{{d}}{{dx}}\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} \\ $$$$\:\:\:\:\:\:+\:\mathrm{6}\:\frac{{d}^{\mathrm{2}} }{{dx}^{\mathrm{2}} }\left(\left({x}−\mathrm{1}\right)\right)^{\mathrm{4}} \:\frac{{d}^{\mathrm{2}} }{{dx}^{\mathrm{2}} }\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} +\mathrm{4}\frac{{d}}{{dx}}\left(\left({x}−\mathrm{1}\right)\right)^{\mathrm{4}} \frac{{d}^{\mathrm{3}} }{{dx}^{\mathrm{3}} }\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} \\ $$$$\:\:\:\:\:\:+\:\mathrm{4}\left({x}−\mathrm{1}\right)^{\mathrm{4}} \frac{{d}^{\mathrm{4}} }{{dx}^{\mathrm{4}} }\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} \\ $$$$=\:\mathrm{24}\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} +\mathrm{96}\left({x}−\mathrm{1}\right)\frac{{d}}{{dx}}\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} +\mathrm{72}\left({x}−\mathrm{1}\right)^{\mathrm{2}} \frac{{d}^{\mathrm{2}} }{{dx}^{\mathrm{2}} }\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} \\ $$$$\:+\:\mathrm{16}\left({x}−\mathrm{1}\right)^{\mathrm{3}} \:\frac{{d}^{\mathrm{3}} }{{dx}^{\mathrm{3}} }\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} +\mathrm{4}\left({x}−\mathrm{1}\right)^{\mathrm{4}} \:\frac{{d}^{\mathrm{4}} }{{dx}^{\mathrm{4}} }\left({x}^{\mathrm{2}} +{x}+\mathrm{5}\right)^{\mathrm{4}} \\ $$

Commented by zakirullah last updated on 03/Nov/21

welldone sir G

$${welldone}\:{sir}\:{G} \\ $$

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