Morrison Wells
05/25/2024 · Elementary School
Differentiate the function. \[ y=(2 x-5)^{4}\left(5-x^{3}\right)^{4} \]
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Tutor-Verified Answer
Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( (2x-5)^4(5-x^3)^4 \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\left(2x-5\right)^{4}\left(5-x^{3}\right)^{4}\right)\)
- step1: Simplify:
\(\frac{d}{dx}\left(\left(10x-2x^{4}-25+5x^{3}\right)^{4}\right)\)
- step2: Use differentiation rules:
\(\frac{d}{dg}\left(g^{4}\right)\times \frac{d}{dx}\left(10x-2x^{4}-25+5x^{3}\right)\)
- step3: Find the derivative:
\(4g^{3}\times \frac{d}{dx}\left(10x-2x^{4}-25+5x^{3}\right)\)
- step4: Calculate:
\(4g^{3}\left(10-8x^{3}+15x^{2}\right)\)
- step5: Substitute back:
\(4\left(10x-2x^{4}-25+5x^{3}\right)^{3}\left(10-8x^{3}+15x^{2}\right)\)
- step6: Calculate:
\(4\left(10375x^{3}-2475x^{6}-7500x^{2}+4500x^{5}+245x^{9}-900x^{8}+18750x-11250x^{4}+2250x^{7}-8x^{12}+60x^{11}-150x^{10}-15625\right)\left(10-8x^{3}+15x^{2}\right)\)
- step7: Multiply the terms:
\(\left(41500x^{3}-9900x^{6}-30000x^{2}+18000x^{5}+980x^{9}-3600x^{8}+75000x-45000x^{4}+9000x^{7}-32x^{12}+240x^{11}-600x^{10}-62500\right)\left(10-8x^{3}+15x^{2}\right)\)
- step8: Apply the distributive property:
\(41500x^{3}\times 10-41500x^{3}\times 8x^{3}+41500x^{3}\times 15x^{2}-9900x^{6}\times 10-\left(-9900x^{6}\times 8x^{3}\right)-9900x^{6}\times 15x^{2}-30000x^{2}\times 10-\left(-30000x^{2}\times 8x^{3}\right)-30000x^{2}\times 15x^{2}+18000x^{5}\times 10-18000x^{5}\times 8x^{3}+18000x^{5}\times 15x^{2}+980x^{9}\times 10-980x^{9}\times 8x^{3}+980x^{9}\times 15x^{2}-3600x^{8}\times 10-\left(-3600x^{8}\times 8x^{3}\right)-3600x^{8}\times 15x^{2}+75000x\times 10-75000x\times 8x^{3}+75000x\times 15x^{2}-45000x^{4}\times 10-\left(-45000x^{4}\times 8x^{3}\right)-45000x^{4}\times 15x^{2}+9000x^{7}\times 10-9000x^{7}\times 8x^{3}+9000x^{7}\times 15x^{2}-32x^{12}\times 10-\left(-32x^{12}\times 8x^{3}\right)-32x^{12}\times 15x^{2}+240x^{11}\times 10-240x^{11}\times 8x^{3}+240x^{11}\times 15x^{2}-600x^{10}\times 10-\left(-600x^{10}\times 8x^{3}\right)-600x^{10}\times 15x^{2}-62500\times 10-\left(-62500\times 8x^{3}\right)-62500\times 15x^{2}\)
- step9: Multiply the numbers:
\(415000x^{3}-332000x^{6}+622500x^{5}-99000x^{6}-\left(-79200x^{9}\right)-148500x^{8}-300000x^{2}-\left(-240000x^{5}\right)-450000x^{4}+180000x^{5}-144000x^{8}+270000x^{7}+9800x^{9}-7840x^{12}+14700x^{11}-36000x^{8}-\left(-28800x^{11}\right)-54000x^{10}+750000x-600000x^{4}+1125000x^{3}-450000x^{4}-\left(-360000x^{7}\right)-675000x^{6}+90000x^{7}-72000x^{10}+135000x^{9}-320x^{12}-\left(-256x^{15}\right)-480x^{14}+2400x^{11}-1920x^{14}+3600x^{13}-6000x^{10}-\left(-4800x^{13}\right)-9000x^{12}-625000-\left(-500000x^{3}\right)-937500x^{2}\)
- step10: Remove the parentheses:
\(415000x^{3}-332000x^{6}+622500x^{5}-99000x^{6}+79200x^{9}-148500x^{8}-300000x^{2}+240000x^{5}-450000x^{4}+180000x^{5}-144000x^{8}+270000x^{7}+9800x^{9}-7840x^{12}+14700x^{11}-36000x^{8}+28800x^{11}-54000x^{10}+750000x-600000x^{4}+1125000x^{3}-450000x^{4}+360000x^{7}-675000x^{6}+90000x^{7}-72000x^{10}+135000x^{9}-320x^{12}+256x^{15}-480x^{14}+2400x^{11}-1920x^{14}+3600x^{13}-6000x^{10}+4800x^{13}-9000x^{12}-625000+500000x^{3}-937500x^{2}\)
- step11: Add the terms:
\(2040000x^{3}-1106000x^{6}+1042500x^{5}+224000x^{9}-328500x^{8}-1237500x^{2}-1500000x^{4}+720000x^{7}-17160x^{12}+45900x^{11}-132000x^{10}+750000x+256x^{15}-2400x^{14}+8400x^{13}-625000\)
The first-order derivative of the function \( y=(2x-5)^4(5-x^3)^4 \) with respect to \( x \) is:
\[ 2040000x^{3}-1106000x^{6}+1042500x^{5}+224000x^{9}-328500x^{8}-1237500x^{2}-1500000x^{4}+720000x^{7}-17160x^{12}+45900x^{11}-132000x^{10}+750000x+256x^{15}-2400x^{14}+8400x^{13}-625000 \]
Quick Answer
The first-order derivative of \( y=(2x-5)^4(5-x^3)^4 \) is \( 2040000x^{3}-1106000x^{6}+1042500x^{5}+224000x^{9}-328500x^{8}-1237500x^{2}-1500000x^{4}+720000x^{7}-17160x^{12}+45900x^{11}-132000x^{10}+750000x+256x^{15}-2400x^{14}+8400x^{13}-625000 \).
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