Find the derivative of 2 + 4 sin ⁡ ( π x 4 ) .

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Find the derivative of       2    +    4    sin    ⁡    (                  π        x            4        )  .

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Use&amp;nbsp;axn=axn&amp;nbsp;to rewrite&amp;nbsp;2+4sin(πx4)&amp;nbsp;as&amp;nbsp;(2+4sin(πx4))12.ddx[(2+4sin(πx4))12]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)=x12&amp;nbsp;and&amp;nbsp;g(x)=2+4sin(πx4).12(2+4sin(πx4))12-1ddx[2+4sin(πx4)]To write&amp;nbsp;-1&amp;nbsp;as a fraction with a common denominator, multiply by&amp;nbsp;22.12(2+4sin(πx4))12-1⋅22ddx[2+4sin(πx4)]Combine&amp;nbsp;-1&amp;nbsp;and&amp;nbsp;22.12(2+4sin(πx4))12+-1⋅22ddx[2+4sin(πx4)]Combine the numerators over the common denominator.12(2+4sin(πx4))1-1⋅22ddx[2+4sin(πx4)]Simplify the numerator.12(2+4sin(πx4))-12ddx[2+4sin(πx4)]Combine fractions.12(2+4sin(πx4))12ddx[2+4sin(πx4)]By the Sum Rule, the derivative of&amp;nbsp;2+4sin(πx4)&amp;nbsp;with respect to&amp;nbsp;x&amp;nbsp;is&amp;nbsp;ddx[2]+ddx[4sin(πx4)].12(2+4sin(πx4))12(ddx[2]+ddx[4sin(πx4)])Since&amp;nbsp;2&amp;nbsp;is constant with respect to&amp;nbsp;x, the derivative of&amp;nbsp;2&amp;nbsp;with respect to&amp;nbsp;x&amp;nbsp;is&amp;nbsp;0.12(2+4sin(πx4))12(0+ddx[4sin(πx4)])Add&amp;nbsp;0&amp;nbsp;and&amp;nbsp;ddx[4sin(πx4)].12(2+4sin(πx4))12ddx[4sin(πx4)]Since&amp;nbsp;4&amp;nbsp;is constant with respect to&amp;nbsp;x, the derivative of&amp;nbsp;4sin(πx4)&amp;nbsp;with respect to&amp;nbsp;x&amp;nbsp;is&amp;nbsp;4ddx[sin(πx4)].12(2+4sin(πx4))12(4ddx[sin(πx4)])Simplify terms.2⋅22(2+4sin(πx4))12ddx[sin(πx4)]Cancel the common factors.2(2+4sin(πx4))12ddx[sin(πx4)]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)=sin(x)&amp;nbsp;and&amp;nbsp;g(x)=πx4.2(2+4sin(πx4))12(cos(πx4)ddx[πx4])Differentiate using the Constant Multiple Rule.2(πcos(πx4))4(2+4sin(πx4))12ddx[x]Cancel the common factors.πcos(πx4)2(2+4sin(πx4))12ddx[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.πcos(πx4)2(2+4sin(πx4))12⋅1Multiply&amp;nbsp;πcos(πx4)2(2+4sin(πx4))12&amp;nbsp;by&amp;nbsp;1.πcos(πx4)2(2+4sin(πx4))12&amp;nbsp;

Find the derivative of <msqrt> 2 + 4 sin ⁡<!-- ⁡ <mro

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