Question

Evaluate $ {n^{th}} $ derivative of $ {e^{ax}} $ .

Answer 1

Hint: Before attempting this question, one should have prior knowledge about the concept of derivative and also remember to identity the pattern of differentiation of the given function and use $ \dfrac{d}{{dx}}{e^x} = {e^x} $ , use this information to approach the solution.

Complete step-by-step answer:
Let $ y = {e^{ax}} $
Since, we have to find the $ {n^{{\text{th}}}} $ derivative of the above given function.
Let us differentiate the given function once with respect to x, we get
First derivative, \[{y_1} = \dfrac{{dy}}{{dx}}\]
\[{y_1} = \dfrac{{d\left( {{e^{ax}}} \right)}}{{dx}}\]
Since we know that $ \dfrac{d}{{dx}}{e^x} = {e^x} $ and $ \dfrac{d}{{dx}}\left( x \right) = 1 $
\[{y_1} = {e^{ax}}\dfrac{d}{{dx}}\left( {ax} \right) = a{e^{ax}}\] (equation 1)
Now, let us differentiate the given function again with respect to x, we get
Second derivative, $ {y_2} = \dfrac{{d{y_1}}}{{dx}} $
 $ {y_2} = \dfrac{d}{{dx}}\left( {a{e^{ax}}} \right) $
 $ {y_2} = a{e^{ax}}\dfrac{d}{{dx}}\left( {ax} \right) $
 $ {y_2} = {a^2}{e^{ax}} $ (equation 2)
Now, let us differentiate the given function again with respect to x, we get
Third derivative, \[{y_3} = \dfrac{{d{y_2}}}{{dx}}\]
\[{y_3} = \dfrac{d}{{dx}}\left( {{a^2}{e^{ax}}} \right)\]
\[{y_3} = {a^2}{e^{ax}}\dfrac{d}{{dx}}\left( {ax} \right)\]
\[{y_3} = {a^3}{e^{ax}}\] (equation 3)
After observing equations (1), (2) and (3), we can say that these derivatives are following a specific pattern and according to this pattern we can write the $ {n^{{\text{th}}}} $ derivative of the given function as
 $ {y_n} = \dfrac{{{d^n}}}{{d{x^n}}}\left( {{e^{ax}}} \right) = {a^n}{e^{ax}} $ .

Note: In these types of problems, the given function is differentiated one by one in order to obtain a pattern which will considerably lead us to the final $ {n^{{\text{th}}}} $ derivative of the given function satisfying that particular pattern.