Question

How do you find the derivative of $e^{-3x}$?

Answer 1

Hint:
Here we need to know that the derivative of ${\displaystyle \frac{d}{dx}}{e}^{ax}=a{e}^{ax}$ which means that the coefficient of the $x$ in the power needs to be noticed and terms with the $x$ in the power of the exponential are also needed to be multiplied with the term which we have got.

Complete step by step solution:
Here we are given to find the derivative of $e^{ax}$
We need to know that when we have the derivative term as $e^x$ then its differentiation is also the same $e^x$.
But here there is not simply the term $e^x$instead we have the term where we also have the coefficient in the power with the variable $x$ so we need to know that when we have the coefficient of degree then that coefficient will also be multiplied with the differentiation of the exponential.
${\displaystyle \frac{d}{dx}}{e}^{ax}=a{e}^{ax}$
So here also we have multiplied the differentiation of the coefficient of the $x$along with the exponential term after the differentiation.
${\displaystyle \frac{d}{dx}}{e}^{ax}=a{e}^{ax}$$---(1)$
Now comparing $e^{ax}$ with$e^{-3x}$, we can write that:
$a=-3$
Now we know that differentiation of ${\displaystyle \frac{d}{dx}}{e}^{ax}=a{e}^{ax}$
Hence we can compare and say:
${\displaystyle \frac{d}{dx}}{e}^{-3x}$
So we will get ${\displaystyle \frac{d}{dx}}{e}^{-3x}=e^{-3x}.{\displaystyle \frac{d}{dx}}\left(-3x\right)$$-----(2)$
We also know that differentiation of ${\displaystyle \frac{d}{dx}}\left(ax\right)=a$
Hence in the similar way we can say that:
${\displaystyle \frac{d}{dx}}\left(-3x\right)=-3$
So now we need to substitute this value in the equation (2) and we will get:
${\displaystyle \frac{d}{dx}}{e}^{-3x}=e^{-3x}.{\displaystyle \frac{d}{dx}}\left(-3x\right)=e^{-3x}.\left(-3\right)$

${\displaystyle \frac{d}{dx}}{e}^{-3x}=-3e^{-3x}$

Note:
Here the student must know that whenever we need to differentiate we also need to multiply with exponential terms the differentiation of the power of $e$.
Similarly when we need to integrate then we need to divide the term instead of multiplication.