Question

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

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.