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If \[y = {e^{\log \left( {{x^5}} \right)}}\] then find the value of the first derivative \[\dfrac{{dy}}{{dx}}\].

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Hint: We will first consider the given function of \[y\]. We need to find the first derivative of the given function. As we know that \[y = {e^{\log x}}\] is equal to \[y = x\], so we will use this concept here and simplify the value of function of \[y\]. Next, find the derivative of the obtained value and hence the result.

Complete step by step solution: First, we will consider the given function \[y = {e^{\log \left( {{x^5}} \right)}}\]
Next, we have to find the derivative \[\dfrac{{dy}}{{dx}}\].
As we know that \[y = {e^{\log x}}\] is equal to \[y = x\].
So, we will use this concept here and simplify the given expression,
Thus, we get,
\[
   \Rightarrow y = {e^{\log \left( {{x^5}} \right)}} \\
   \Rightarrow y = {x^5} \\
 \]
Now, we will find the derivative of the obtained expression by differentiating \[y\] with respect to \[x\].
Thus, we get,
\[
   \Rightarrow \dfrac{{dy}}{{dx}} = 5{x^{5 - 1}} \\
   \Rightarrow \dfrac{{dy}}{{dx}} = 5{x^4} \\
 \]
Here, we have applied the formula of differentiation that is, \[\dfrac{{dy}}{{dx}} = n{x^{n - 1}}\].

Thus, the derivative of the given function is \[\dfrac{{dy}}{{dx}} = 5{x^4}\].

Note: Note: Logarithmic functions are the inverses of exponential functions. We have used the formula of differentiation \[\dfrac{{dy}}{{dx}} = n{x^{n - 1}}\] to find the derivative of the function. The exponential and logarithmic functions are inverse of each other. Differentiate the variable \[y\] with respect to \[x\] to find the derivative.