Differentiating rational functions
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Exercise:
nprvmulticols abclist abc displaystyle fracddddx left fracx right abc displaystyle fracddddx left fracx^ right abc displaystyle fracddddx left fracxx + right %d abc displaystyle fracddddx left fracx^x + right abc displaystyle fracddddx left fracx^ + x - right abc displaystyle fracddddx left fracx + x^ + right abc displaystyle fracddddx left fracx^ + x + x^ + right %h abc displaystyle fracddddx left fracx^x^ + right %i abc displaystyle fracddddx left fracx^ - x^ + right %j abc displaystyle fracddddx left fracx^ + x + right abc displaystyle fracddddx left fracx^ + x^ + right abc displaystyle fracddddx left fracx^ + xx^ + right abc displaystyle fracddddx left fracx + x^ - right abc displaystyle fracddddx left fracx^x^ + x + right abc displaystyle fracddddx left fracx^ + x + x^ - x + right abc displaystyle fracddddx left fracx^x^ + right abc displaystyle fracddddx left fracx^ + x^ + x^ + right abc displaystyle fracddddx left fracx^ + x^ + x^ + x + right abc displaystyle fracddddx left fracx^x - x^ + x + right abc displaystyle fracddddx left fracx - ^x + ^ right abc displaystyle fracddddx left fracxx + right abc displaystyle fracddddx left fracx^ + xx - right abc displaystyle fracddddx left fracx^ + ^x^ right abc displaystyle fracddddx left fracx^ + x^ + x + x^ + right abclist nprvmulticols
Solution:
abclist abc Use polynom rule: fracddddx x^n nx^n- * fracddddx left fracx right -fracx^ * abc Use polynom rule: fracddddx x^n nx^n- * fracddddx left fracx^ right -fracx^ * %c abc Use quotient rule: leftfracfgright^- fracf'g-fg'g^ * fracddddx left fracxx + right fracx + - xx + ^ fracx + ^ * %d abc Use quotient rule: leftfracfgright^- fracf'g-fg'g^ * fracddddx left fracx^x + right fracxx + - x^x + ^ fracxx + x + ^ * %e abc * fracddddx left fracx^x + right fracxx + - x^x + ^ fracxx - - x^ + x - ^ * %f abc * fracddddx left fracx + x^ + right fracx^ + - x + xx^ + ^ * %g abc * fracddddx left fracx^ + x + x^ + right fracx + x^ + - x^ + x + xx^ + ^ * %h abc * fracddddx left fracx^x^ + right fracx^x^ + - x^xx^ + ^ * %i abc * fracddddx left fracx^ - x^ + right fracxx^ + - x^ - xx^ + ^ * %j abc * fracddddx left fracx^ + x + right frac- x + x^ + x + ^ * %k abc * fracddddx left fracx^ + x^ + right fracxx^ + - x^ + x^x^ + ^ * %l abc fracx^ + x^ + - x^ + xxx^ + ^ abc fracx^ - - x + xx^ - ^ abc fracx^x^ + x + - x^x + x^ + x + ^ abc Quotient rule: numerator x^ + x^ - x + - x^ + x + x - denominator: x^ - x + ^ abc fracx^x^ + - x^x^x^ + ^ abc fracxx^ + x^ + - x^ + x^ + xx^ + x^ + ^ abc abc Expand numerator first if desired: x^ - x^ Then quotient rule over denominator x + ^ abc fracx - x + ^ - x - ^ x + x + ^ abc frac- x + - x xx + ^ frac-x + + xx^x + ^ abc abc abc abclist
nprvmulticols abclist abc displaystyle fracddddx left fracx right abc displaystyle fracddddx left fracx^ right abc displaystyle fracddddx left fracxx + right %d abc displaystyle fracddddx left fracx^x + right abc displaystyle fracddddx left fracx^ + x - right abc displaystyle fracddddx left fracx + x^ + right abc displaystyle fracddddx left fracx^ + x + x^ + right %h abc displaystyle fracddddx left fracx^x^ + right %i abc displaystyle fracddddx left fracx^ - x^ + right %j abc displaystyle fracddddx left fracx^ + x + right abc displaystyle fracddddx left fracx^ + x^ + right abc displaystyle fracddddx left fracx^ + xx^ + right abc displaystyle fracddddx left fracx + x^ - right abc displaystyle fracddddx left fracx^x^ + x + right abc displaystyle fracddddx left fracx^ + x + x^ - x + right abc displaystyle fracddddx left fracx^x^ + right abc displaystyle fracddddx left fracx^ + x^ + x^ + right abc displaystyle fracddddx left fracx^ + x^ + x^ + x + right abc displaystyle fracddddx left fracx^x - x^ + x + right abc displaystyle fracddddx left fracx - ^x + ^ right abc displaystyle fracddddx left fracxx + right abc displaystyle fracddddx left fracx^ + xx - right abc displaystyle fracddddx left fracx^ + ^x^ right abc displaystyle fracddddx left fracx^ + x^ + x + x^ + right abclist nprvmulticols
Solution:
abclist abc Use polynom rule: fracddddx x^n nx^n- * fracddddx left fracx right -fracx^ * abc Use polynom rule: fracddddx x^n nx^n- * fracddddx left fracx^ right -fracx^ * %c abc Use quotient rule: leftfracfgright^- fracf'g-fg'g^ * fracddddx left fracxx + right fracx + - xx + ^ fracx + ^ * %d abc Use quotient rule: leftfracfgright^- fracf'g-fg'g^ * fracddddx left fracx^x + right fracxx + - x^x + ^ fracxx + x + ^ * %e abc * fracddddx left fracx^x + right fracxx + - x^x + ^ fracxx - - x^ + x - ^ * %f abc * fracddddx left fracx + x^ + right fracx^ + - x + xx^ + ^ * %g abc * fracddddx left fracx^ + x + x^ + right fracx + x^ + - x^ + x + xx^ + ^ * %h abc * fracddddx left fracx^x^ + right fracx^x^ + - x^xx^ + ^ * %i abc * fracddddx left fracx^ - x^ + right fracxx^ + - x^ - xx^ + ^ * %j abc * fracddddx left fracx^ + x + right frac- x + x^ + x + ^ * %k abc * fracddddx left fracx^ + x^ + right fracxx^ + - x^ + x^x^ + ^ * %l abc fracx^ + x^ + - x^ + xxx^ + ^ abc fracx^ - - x + xx^ - ^ abc fracx^x^ + x + - x^x + x^ + x + ^ abc Quotient rule: numerator x^ + x^ - x + - x^ + x + x - denominator: x^ - x + ^ abc fracx^x^ + - x^x^x^ + ^ abc fracxx^ + x^ + - x^ + x^ + xx^ + x^ + ^ abc abc Expand numerator first if desired: x^ - x^ Then quotient rule over denominator x + ^ abc fracx - x + ^ - x - ^ x + x + ^ abc frac- x + - x xx + ^ frac-x + + xx^x + ^ abc abc abc abclist