Question

Find the derivatives w.r.t.x:
$\rm \frac{e^{x}}{1+x^{5}}$

Answer 1

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Answer 2

Answer:

$\rm \dfrac{d}{dx}\left ( \dfrac {e^x}{1+x^5}\right)=\dfrac{e^x(1+x^5-5x^4)}{(1+x^5)^2}.$

Step-by-step explanation:

Given function:

$\rm \dfrac{e^{x}}{1+x^{5}}$

The derivative of a function which is in fraction, such as, $\rm \dfrac{f(x)}{g(x)}$ is given by

$\rm \dfrac{d}{dx}\left ( \dfrac {f(x)}{g(x)}\right) = \dfrac{g(x)\dfrac{d}{dx}(f(x))-f(x)\dfrac{d}{dx}(g(x))}{g^2(x)}$.

Here, we have,

$\rm f(x)=e^x\\ g(x)=1+x^5$.

Therefore,

$\rm \dfrac{d}{dx}\left ( \dfrac {e^x}{1+x^5}\right) = \dfrac{(1+x^5)\dfrac{d}{dx}(e^x)-e^x\dfrac{d}{dx}(1+x^5)}{(1+x^5)^2}\\=\dfrac{(1+x^5)e^x-e^x(5x^4)}{(1+x^5)^2}\\=\dfrac{e^x(1+x^5-5x^4)}{(1+x^5)^2}.$