attempt 1: 10 attempts remaining. if $f(x)=10e^{x}-4x^{4}+1$, find $f(x)$. $f(x)=$

Explanation:

Step1: Differentiate term - by - term

Use the sum/difference rule of differentiation $(u\pm v\pm w)' = u'\pm v'\pm w'$. Here $u = 10e^{x}$, $v = 4x^{4}$, $w = 1$.

Step2: Differentiate $10e^{x}$

The derivative of $e^{x}$ is $e^{x}$, and by the constant - multiple rule $(cf(x))'=cf'(x)$ where $c = 10$, so $(10e^{x})'=10e^{x}$.

Step3: Differentiate $4x^{4}$

By the power rule $(x^{n})'=nx^{n - 1}$, for $n = 4$ and $c = 4$, we have $(4x^{4})'=4\times4x^{4 - 1}=16x^{3}$.

Step4: Differentiate the constant 1

The derivative of a constant $c$ is 0, so $(1)' = 0$.

Step5: Combine the derivatives

$f'(x)=(10e^{x})'-(4x^{4})'+(1)'=10e^{x}-16x^{3}+0$.

Answer:

$10e^{x}-16x^{3}$