Question

differentiate
f(x)=e^{x^{5}} + 4x
f(x)=□

Explanation:

Step1: Apply sum - rule of differentiation

The sum - rule states that if $y = u + v$, then $y'=u'+v'$. Here, $u = e^{x^{5}}$ and $v = 4x$. So, $f'(x)=\frac{d}{dx}(e^{x^{5}})+\frac{d}{dx}(4x)$.

Step2: Differentiate $4x$

Using the power - rule $\frac{d}{dx}(ax)=a$ (where $a$ is a constant), for $v = 4x$, $\frac{d}{dx}(4x)=4$.

Step3: Differentiate $e^{x^{5}}$ using chain - rule

The chain - rule states that if $y = f(g(x))$, then $y'=f'(g(x))\cdot g'(x)$. Let $g(x)=x^{5}$ and $f(u)=e^{u}$, where $u = x^{5}$. First, $\frac{d}{du}(e^{u})=e^{u}$ and $\frac{d}{dx}(x^{5}) = 5x^{4}$. Then $\frac{d}{dx}(e^{x^{5}})=e^{x^{5}}\cdot5x^{4}=5x^{4}e^{x^{5}}$.

Step4: Combine the results

$f'(x)=5x^{4}e^{x^{5}}+4$.

Answer:

$5x^{4}e^{x^{5}} + 4$