Question

How do you find the derivative of $y={{e}^{2x}}$?

Answer 1

To get the derivative of $y={{e}^{2x}}$ with respect to $x$. Firstly, suppose $t=2x$ and get the derivative of $t$ with respect to $x$ . Now we can write $y={{e}^{2x}}$ as $y={{e}^{t}}$ and after that try to get the derivative with respect to $t$. After combining both the derivative as $\dfrac{dy}{dx}=\dfrac{dy}{dt}\times \dfrac{dt}{dx}$ we can get the derivative of $y={{e}^{2x}}$with respect to $x$ .

Complete step by step solution:
The question has the given equation as $y={{e}^{2x}}$
Since, we cannot derive the given equation directly. So, we would assume
$t=2x$ … $\left( i \right)$
After that we have to derive equation $\left( i \right)$ with respect to $x$ as
$\Rightarrow \dfrac{dt}{dx}=\dfrac{d\left( 2x \right)}{dx}$
Since, numbers are constant in any derivation, so we cannot derive$2$ . For$x$, the derivation will be $1$ .
So,
$\Rightarrow \dfrac{dt}{dx}=2$ … $\left( ii \right)$
With the use of equation $\left( i \right)$ , we can write the given equation in the question $y={{e}^{2x}}$ as:
$\Rightarrow y={{e}^{t}}$
Since, the derivation of ${{e}^{t}}$ with respect to $t$ is itself ${{e}^{t}}$ . So, after derivation of the above equation with respect to $t$ will be as
$\Rightarrow \dfrac{dy}{dt}={{e}^{t}}$
Now, after using equation $\left( i \right)$ , we can write the above derivation in term of $x$ as
$\Rightarrow \dfrac{dy}{dt}=$ ${{e}^{2x}}$ … $\left( iii \right)$
Now, for getting the derivative of $y={{e}^{2x}}$ with respect to $x$ , we can use the method
$\dfrac{dy}{dx}=\dfrac{dy}{dt}\times \dfrac{dt}{dx}$ … $\left( iv \right)$
After applying the equation $\left( ii \right)$ and $\left( iii \right)$ in equation $\left( iv \right)$ , we get
$\Rightarrow \dfrac{dy}{dx}={{e}^{2x}}\times 2$
We can write the above equation as
$\Rightarrow \dfrac{dy}{dx}=2{{e}^{2x}}$
Hence, the derivative of the equation $y={{e}^{2x}}$ is $2{{e}^{2x}}$ .

Note:
Here we can check whether the derivative of the given equation is correct or not in the following way-
From the solution, we have:
$\dfrac{dy}{dx}=2{{e}^{2x}}$
We can write it as:
$\Rightarrow dy=\left( 2{{e}^{2x}} \right)dx$
After applying the symbol of integration both sides:
$\Rightarrow \int{{}}dy=\int{\left( 2{{e}^{2x}} \right)}dx$
After integrating the above equation, we will get:
$\begin{aligned} & \Rightarrow y=\dfrac{2{{e}^{2x}}}{2} \\\ & \Rightarrow y={{e}^{2x}} \\\ \end{aligned}$
Now, we got the given equation of the question from the integration of the solution. Hence, the solution is correct.