Find the derivative of 3 + 4 tan ⁡ ( π x ) 3 .

Question

Find the derivative of             3      +      4      tan      ⁡      (      π      x      )        3  .

Ariella Bruce · Accepted Answer

Use&amp;nbsp;axn=axn&amp;nbsp;to rewrite&amp;nbsp;3+4tan(πx)3&amp;nbsp;as&amp;nbsp;(3+4tan(πx))13.ddx[(3+4tan(πx))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)=3+4tan(πx).13(3+4tan(πx))13-1ddx[3+4tan(πx)]To write&amp;nbsp;-1&amp;nbsp;as a fraction with a common denominator, multiply by&amp;nbsp;33.13(3+4tan(πx))13-1⋅33ddx[3+4tan(πx)]Combine&amp;nbsp;-1&amp;nbsp;and&amp;nbsp;33.13(3+4tan(πx))13+-1⋅33ddx[3+4tan(πx)]Combine the numerators over the common denominator.13(3+4tan(πx))1-1⋅33ddx[3+4tan(πx)]Simplify the numerator.13(3+4tan(πx))-23ddx[3+4tan(πx)]Differentiate.43(3+4tan(πx))23ddx[tan(πx)]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)=tan(x)&amp;nbsp;and&amp;nbsp;g(x)=πx.43(3+4tan(πx))23(sec2(πx)ddx[πx])Differentiate.4πsec2(πx)3(3+4tan(πx))23

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

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