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Question: How do you find the derivative of \({e^{ - 3x}}\)?...

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

Explanation

Solution

Here we need to know that the derivative of ddxeax=aeax\dfrac{d}{{dx}}{e^{ax}} = a{e^{ax}} which means that the coefficient of the xx in the power needs to be noticed and terms with the xx 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 eax{e^{ax}}
We need to know that when we have the derivative term as ex{e^x} then its differentiation is also the same ex{e^x}.
But here there is not simply the term ex{e^x}instead we have the term where we also have the coefficient in the power with the variable xx 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.
ddxeax=aeax\dfrac{d}{{dx}}{e^{ax}} = a{e^{ax}}
So here also we have multiplied the differentiation of the coefficient of the xxalong with the exponential term after the differentiation.
ddxeax=aeax\dfrac{d}{{dx}}{e^{ax}} = a{e^{ax}} −−−(1) - - - (1)
Now comparing eax{e^{ax}} withe−3x{e^{ - 3x}}, we can write that:
a=−3a = - 3
Now we know that differentiation of ddxeax=aeax\dfrac{d}{{dx}}{e^{ax}} = a{e^{ax}}
Hence we can compare and say:
ddxe−3x\dfrac{d}{{dx}}{e^{ - 3x}}
So we will get ddxe−3x=e−3x.ddx(−3x)\dfrac{d}{{dx}}{e^{ - 3x}} = {e^{ - 3x}}.\dfrac{d}{{dx}}\left( { - 3x} \right) −−−−−(2) - - - - - (2)
We also know that differentiation of ddx(ax)=a\dfrac{d}{{dx}}\left( {ax} \right) = a
Hence in the similar way we can say that:
ddx(−3x)=−3\dfrac{d}{{dx}}\left( { - 3x} \right) = - 3
So now we need to substitute this value in the equation (2) and we will get:
ddxe−3x=e−3x.ddx(−3x)=e−3x.(−3)\dfrac{d}{{dx}}{e^{ - 3x}} = {e^{ - 3x}}.\dfrac{d}{{dx}}\left( { - 3x} \right) = {e^{ - 3x}}.\left( { - 3} \right)

ddxe−3x=−3e−3x\dfrac{d}{{dx}}{e^{ - 3x}} = - 3{e^{ - 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 ee.
Similarly when we need to integrate then we need to divide the term instead of multiplication.