Question

How do you find the derivative of ${{e}^{\sqrt{x}}}$?

Answer 1

Hint: In this problem we need to calculate the derivative of the given function. We will first assume the given function is equal to any variable say $y$ and note it as equation one. We can observe that the given function is an exponential function, so we will apply natural logarithm on both sides of the above equation and simplify the equation by using the logarithmic formulas. Now we will differentiate the obtained equation with respect to $x$ and simplify the equation by differentiation formulas. Now we will use the value of $y$ from the equation one to get the final result.

Complete step-by-step solution:
Given function, ${{e}^{\sqrt{x}}}$.
Let us assume the equation $y={{e}^{\sqrt{x}}}....\left( \text{i} \right)$.
Applying the natural logarithm on both sides of the above equation, then we will get
$\ln y=\ln {{e}^{\sqrt{x}}}$
We have the logarithmic formula $\ln {{e}^{a}}=a$, then we will get
$\ln y=\sqrt{x}$
Differentiating the above equation with respect to $x$, then we will get
$\dfrac{d}{dx}\left( \ln y \right)=\dfrac{d}{dx}\left( \sqrt{x} \right)$
We have the differentiation formulas $\dfrac{d}{dx}\left( \ln x \right)=\dfrac{1}{x}$, $\dfrac{d}{dx}\left( \sqrt{x} \right)=\dfrac{1}{2\sqrt{x}}$. Using these formulas in the above equation, then we will get
$\Rightarrow \dfrac{1}{y}\dfrac{dy}{dx}=\dfrac{1}{2\sqrt{x}}$
Simplifying the above equation, then we will have
$\dfrac{dy}{dx}=\dfrac{y}{2\sqrt{x}}$
Substituting the value of $y$ from equation $\left( \text{i} \right)$, then we will get
$\therefore \dfrac{dy}{dx}=\dfrac{{{e}^{\sqrt{x}}}}{2\sqrt{x}}$

Note: For this problem we can use another two methods. The first one is the substitution method. In this method we will take the substitution and use its differentiation value to get the final result. For this problem we will use the following equation $\sqrt{x}=u\Rightarrow \dfrac{du}{dx}=\dfrac{1}{2\sqrt{x}}$, by using this equation we will get the derivative of the given equation. Now the second method is direct method, we can directly calculate the derivative by following steps
$\begin{align}
  & \dfrac{d}{dx}\left( {{e}^{\sqrt{x}}} \right)={{e}^{\sqrt{x}}}\dfrac{d}{dx}\left( \sqrt{x} \right) \\
 & \Rightarrow \dfrac{d}{dx}\left( {{e}^{\sqrt{x}}} \right)=\dfrac{{{e}^{\sqrt{x}}}}{2\sqrt{x}} \\
\end{align}$
From all the methods we will get the same result.