Question

How do you find the derivative of ${{e}^{\dfrac{1}{x}}}$ ?

Answer 1

As we see the given function, we know the derivative of exponential function if there is $x$ instead of $\dfrac{1}{x}$ so in this situation we use the chain rule according to which we can assume that $\dfrac{1}{x}$ an other variable now that function is easily differentiable with respect to that variable and then multiply it with the differentiation of that assumed term w.r.t. $x$.

Complete step-by-step answer:
As we have to find the derivative of ${{e}^{\dfrac{1}{x}}}$ therefore using chain rule-
Let assume $y={{e}^{\dfrac{1}{x}}}$ and $\dfrac{1}{x}$ be $t$
$\Rightarrow \dfrac{1}{x}=t\Rightarrow y={{e}^{t}}$
Now $y$ is a clear exponential function of linear power of $t$
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dt}\left( {{e}^{t}} \right).\dfrac{dt}{dx}$
$\because \dfrac{d}{dt}\left( {{e}^{t}} \right)={{e}^{t}}$ and $t=\dfrac{1}{x}$
$={{e}^{t}}.\dfrac{d}{dx}\left( \dfrac{1}{x} \right)$
Since $\dfrac{d\left( {{x}^{n}} \right)}{dx}=n.{{x}^{n-1}}$ and substituting $t=\dfrac{1}{x}$
$\Rightarrow {{e}^{\dfrac{1}{x}}}.\left( \dfrac{-1}{{{x}^{2}}} \right)$
Hence, the derivative of ${{e}^{\dfrac{1}{x}}}$ with respect to $x$ is $\dfrac{-{{e}^{\dfrac{1}{x}}}}{{{x}^{2}}}$

Note: First see the function properly that we want to differentiate, and when we know the derivative of that function if there is slightly different variable term then we use the chain rule by assuming that term as another variable and then differentiate the assumed term with the original variable and then multiply both the derivative.