2.1 Let us differentiate f(x)=(2x3+3x2)(x2+5x3+5) using product rule:
f′(x)=(6x2+6x)(x2+5x3+5)+(2x3+3x2)(2x+15x2)=6x4+30x5+30x2+6x3+30x4+30x+4x4+30x5+6x3+45x4=60x5+85x4+12x3+30x2+30x
2.2 Let us differentiate f(x)=(8x3+4x)(2x2+5) using product rule:
f′(x)=(24x2+4)(2x2+5)+(8x3+4x)4x=48x4+120x2+8x2+20+32x4+16x2=80x4+144x2+20
2.3 Let us differentiate f(x)=2x2−4x+55x3+7x using quotient rule:
f′(x)=(2x2−4x+5)2(15x2+7)(2x2−4x+5)−(5x3+7x)(4x−4)=(2x2−4x+5)230x4−60x3+75x2+14x2−28x+35−20x4+20x3−28x2+28x=(2x2−4x+5)210x4−40x3+61x2+35
2.4 Let us differentiate f(x)=4x−5x3+4 using quotient rule:
f′(x)=(4x−5)23x2(4x−5)−(x3+4)4=(4x−5)28x3−15x2−16
2.5 Let us differentiate f(x)=(x5+6)5 using chain rule:
f′(x)=5(x5+6)4(x5+6)′=25x4(x5+6)4.
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