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How do you differentiate \[{e^{ - 10x}}\] ?

Last updated date: 01st Mar 2024
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IVSAT 2024
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Hint: In this question, we are given an exponential function and we have to find its derivative, the function involves e raised to the power -10x, so we have to differentiate \[{e^{ - 10x}}\] with respect to x. We will first differentiate the whole quantity \[{e^{ - 10x}}\] and then differentiate the quantity that is written in the power $( - 10x)$ as it is also a function of x. The result of multiplying these two differentiated functions will give the value of $\dfrac{{dy}}{{dx}}$ or $y'(x)$ . On solving the given question using the above information, we will get the correct answer.

Complete step-by-step solution:
We have to differentiate \[{e^{ - 10x}}\]
Let $y = {e^{ - 10x}}$
We know that $\dfrac{{d{e^x}}}{{dx}} = {e^x}$
So differentiating both sides of the above equation with respect to x, we get –
$\dfrac{{dy}}{{dx}} = {e^{ - 10x}}\dfrac{{d( - 10x)}}{{dx}}$
We also know that $\dfrac{{dkx}}{{dx}} = kx$ , so we get –
$\dfrac{{dy}}{{dx}} = - 10{e^{ - 10x}}$
Hence, the derivative of \[{e^{ - 10x}}\] is $ - 10{e^{ - 10x}}$ .

Note: Differentiation is represented as $\dfrac{{dy}}{{dx}}$ and is used when we have to find the instantaneous rate of change of a quantity. In the expression $\dfrac{{dy}}{{dx}}$ , $dy$ represents a very small change in quantity and $dx$ represents the small change in the quantity with respect to which the given quantity is changing.
In this question, we have to differentiate \[{e^{ - 10x}}\] , it is a function containing only one variable quantity, so we can simply start differentiating it. But we must rearrange the equation if the equation contains more than one variable quantity so that the variable with respect to which the function is differentiated is present on one side and the variable whose derivative we have to find is present on the other side.
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