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Question

# Calculate the derivatives of the functions. $f(x)=\left[(6.4 x-1)^{2}+(5.4 x-2)^{3}\right]^{2}$

Solution

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$\begin{gathered} f\left( x \right) = {\left[ {{{\left( {6.4x - 1} \right)}^2} + {{\left( {5.4x - 2} \right)}^3}} \right]^2} \\ \textcolor{#4257b2}{{\text{Differentiate both sides with respect to }}x} \\ f'\left( x \right) = \frac{d}{{dx}}{\left[ {{{\left( {6.4x - 1} \right)}^2} + {{\left( {5.4x - 2} \right)}^3}} \right]^2} \\ \textcolor{#4257b2}{ {\text{Apply the chain rule}}} \\ f'\left( x \right) = 2{\left[ {{{\left( {6.4x - 1} \right)}^2} + {{\left( {5.4x - 2} \right)}^3}} \right]^{2 - 1}}\frac{d}{{dx}}\left[ {{{\left( {6.4x - 1} \right)}^2} + {{\left( {5.4x - 2} \right)}^3}} \right] \\ \textcolor{#4257b2}{ {\text{Therefore}}{\text{,}}} \\ f'\left( x \right) = 2\left[ {{{\left( {6.4x - 1} \right)}^2} + {{\left( {5.4x - 2} \right)}^3}} \right]\left[ {2\left( {6.4x - 1} \right)\left( {6.4} \right) + 3{{\left( {5.4x - 2} \right)}^2}\left( {5.4} \right)} \right] \\ \textcolor{#4257b2}{ {\text{Simplify}}} \\ f'\left( x \right) = 2\left[ {{{\left( {6.4x - 1} \right)}^2} + {{\left( {5.4x - 2} \right)}^3}} \right]\left[ {12.8\left( {6.4x - 1} \right) + 16.2{{\left( {5.4x - 2} \right)}^2}} \right] \\ f'\left( x \right) = \left[ {25.6\left( {6.4x - 1} \right) + 32.4{{\left( {5.4x - 2} \right)}^2}} \right]\left[ {{{\left( {6.4x - 1} \right)}^2} + {{\left( {5.4x - 2} \right)}^3}} \right] \\ \end{gathered}$

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