Derivative of Power of a Function

To find a derivative of a function, consider the examples to elaborate the concept of derivative of a given function as
1. Consider an example of function

y =  {\left( {2{x^2} + 4x + 10} \right)^2}


Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}}  = \frac{d}{{dx}}{\left( {2{x^2} + 4x + 10} \right)^2}


First we take the derivative of power of a function, and then differentiate the inner function

\begin{gathered}\frac{{dy}}{{dx}} = 2{\left( {2{x^2} + 4x +  10} \right)^{2 - 1}}\frac{d}{{dx}}\left( {2{x^2} + 4x + 10} \right) \\ \frac{{dy}}{{dx}} = 2\left( {2{x^2} + 4x +  10} \right)\left( {2{x^2} + 4} \right) \\ \end{gathered}

2. Consider an example of function

y =  {\left( {5x + 8} \right)^4}


Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}}  = \frac{d}{{dx}}{\left( {5x + 8} \right)^4}


First we take the derivative of power of a function, and then differentiate the inner function

\begin{gathered}\frac{{dy}}{{dx}} = 4{\left( {5x + 8}  \right)^{4 - 1}}\frac{d}{{dx}}\left( {5x + 8} \right) \\ \frac{{dy}}{{dx}} = 20{\left( {5x + 8}  \right)^3} \\ \end{gathered}

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