Find the derivative of ln ⁡ ( 3 + 4 6 + x 3 ).

Question

Find the derivative of   ln  ⁡  (  3  +  4            6      +      x        3    ).

Lucille Melton · Accepted Answer

Use&amp;nbsp;axn=axn&amp;nbsp;to rewrite&amp;nbsp;6+x3&amp;nbsp;as&amp;nbsp;(6+x)13.ddx[ln(3+4(6+x)13)]Differentiate using the chain rule, which states that&amp;nbsp;ddx[f(g(x))]&amp;nbsp;is&amp;nbsp;f′(g(x))g′(x)&amp;nbsp;where&amp;nbsp;f(x)=ln(x)&amp;nbsp;and&amp;nbsp;g(x)=3+4(6+x)13.13+4(6+x)13ddx[3+4(6+x)13]Differentiate.43+4(6+x)13ddx[(6+x)13]Differentiate using the chain rule, which states that&amp;nbsp;ddx[f(g(x))]&amp;nbsp;is&amp;nbsp;f′(g(x))g′(x)&amp;nbsp;where&amp;nbsp;f(x)=x13&amp;nbsp;and&amp;nbsp;g(x)=6+x.43+4(6+x)13(13(6+x)13-1ddx[6+x])To write&amp;nbsp;-1&amp;nbsp;as a fraction with a common denominator, multiply by&amp;nbsp;33.43+4(6+x)13(13(6+x)13-1⋅33ddx[6+x])Combine&amp;nbsp;-1&amp;nbsp;and&amp;nbsp;33.43+4(6+x)13(13(6+x)13+-1⋅33ddx[6+x])Combine the numerators over the common denominator.43+4(6+x)13(13(6+x)1-1⋅33ddx[6+x])Simplify the numerator.43+4(6+x)13(13(6+x)-23ddx[6+x])Combine fractions.43(6+x)23(3+4(6+x)13)ddx[6+x]By the Sum Rule, the derivative of&amp;nbsp;6+x&amp;nbsp;with respect to&amp;nbsp;x&amp;nbsp;is&amp;nbsp;ddx[6]+ddx[x].43(6+x)23(3+4(6+x)13)(ddx[6]+ddx[x])Since&amp;nbsp;6&amp;nbsp;is constant with respect to&amp;nbsp;x, the derivative of&amp;nbsp;6&amp;nbsp;with respect to&amp;nbsp;x&amp;nbsp;is&amp;nbsp;0.43(6+x)23(3+4(6+x)13)(0+ddx[x])Add&amp;nbsp;0&amp;nbsp;and&amp;nbsp;ddx[x].43(6+x)23(3+4(6+x)13)ddx[x]Differentiate using the Power Rule which states that&amp;nbsp;ddx[xn]&amp;nbsp;is&amp;nbsp;nxn-1&amp;nbsp;where&amp;nbsp;n=1.43(6+x)23(3+4(6+x)13)⋅1Multiply&amp;nbsp;43(6+x)23(3+4(6+x)13)&amp;nbsp;by&amp;nbsp;1.43(6+x)23(3+4(6+x)13)&amp;nbsp;

Find the derivative of ln ⁡<!-- ⁡ 3 + 4 <mroot> <mrow> 6 +

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