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How do you differentiate $f\left( x \right) = {e^{3x}}$ ?

Last updated date: 22nd Feb 2024
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IVSAT 2024
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Hint: Use the chain rule to do the differentiation of the function in the question. Now, find the way to do the differentiation of the exponential function. In the chain rule, each function is differentiated one by one and then multiplied with each other.

Complete step by step solution:
Differentiation and integration are the two mathematical concepts which when composed from calculus and both are opposite of each other. Differentiation can be defined as an instantaneous rate of change of the function with respect to one of its variables. It is the change in the value of function due to a change in the independent variable. If a function $y$ is differentiated with respect to $x$, then any change in the value of $y$ due to the change in the value of $x$ is given by –
In the question, we have given the function ${e^{3x}}$, so, we have to know the rule on how to differentiate the exponential function. The derivative of the exponential function is the function itself.
Therefore, we know that –
$\dfrac{d}{{dx}}\left( {{e^x}} \right) = {e^x}$
Now, we have to use the chain rule of differentiation to solve the function ${e^{3x}}$ -
Therefore, differentiating $f\left( x \right)$ with respect to $x$ , we get –
$ \Rightarrow f'\left( x \right) = \dfrac{d}{{dx}}\left( {{e^{3x}}} \right)$
Chain rule involves the differentiation of the function one by one multiplied with each other –
$ \Rightarrow f'\left( x \right) = {e^{3x}} \times \dfrac{d}{{dx}}\left( {3x} \right)$
Now, differentiating $3x$ with respect to $x$, we know that differentiation of $x$ is 1, so, we get –
   \Rightarrow f'\left( x \right) = {e^{3x}}.3 \\
   \Rightarrow f'\left( x \right) = 3{e^{3x}} \\

Hence, the differentiation of ${e^{3x}}$ is $3{e^{3x}}$.

The integration is the opposite of the differentiation, so, if we integrate the function $3{e^{3x}}$ with respect to $x$ we will again get back the function which is written in the question, ${e^{3x}}$. We usually do the integration by using the substitution method.