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Explanation:Given:
[tex]y\text{ = \lparen5x}^2-\text{ 8x + 3\rparen}^5[/tex]To find:
derivative of the function = dy/dx
To determine the derivative, we will follow the steps provided in the question
[tex]\begin{gathered} a)\text{ }let\text{ the inside = expression in the parenthesis} \\ \text{inside: u = 5x}^2\text{ - 8x + 3} \\ \\ Substitute\text{ for u} \\ y\text{ = u}^5 \\ outside:\text{ y = u}^5 \end{gathered}[/tex]b) Next is to find the derivative of the outside and insides:
[tex]\begin{gathered} inside:\text{ u = 5x}^2\text{ - 8x + 3} \\ \frac{du}{dx}\text{ = 10x - 8} \\ \\ outside:\text{ y = u}^5 \\ \frac{dy}{du}\text{ = 5u}^{5-1} \\ \frac{dy}{du}\text{ = 5u}^4 \end{gathered}[/tex]c) We will use the chain rule formula to get the derivative:
[tex]\begin{gathered} \frac{dy}{dx}=\text{ }\frac{dy}{du}\times\frac{du}{dx} \\ \\ \frac{dy}{dx}=\text{ 5u}^4\text{ }\times\text{ \lparen10x - 8\rparen} \\ \\ recall\text{ u = 5x}^2\text{ - 8x + 3} \\ substitute\text{ for u in the derivative above:} \\ \frac{dy}{dx}=\text{ 5\lparen5x}^2-\text{ 8x + 3\rparen}^4\text{ }\times\text{ \lparen10x - 8\rparen} \end{gathered}[/tex][tex]\frac{dy}{dx}=\text{ 5\lparen10x - 8\rparen\lparen5x}^2\text{ - 8x + 3\rparen}^4[/tex]