Question

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

Answer 1

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.