Question

hw_2.1_basic derivative rules
due sep 14, 2025 11:59 pm.
11.
if $f(t)=6t^{4}-4t + 4e^{t}$, find $f(t)$.
$f(t)=$

Explanation:

Step1: Apply power - rule to $6t^4$

The power - rule states that if $y = ax^n$, then $y'=anx^{n - 1}$. For $y = 6t^4$, $a = 6$ and $n = 4$. So the derivative is $6\times4t^{4 - 1}=24t^3$.

Step2: Apply power - rule to $-4t$

For $y=-4t$, where $a=-4$ and $n = 1$. Using the power - rule $y'=-4\times1t^{1 - 1}=-4$.

Step3: Apply derivative rule for exponential function to $4e^t$

The derivative of $e^x$ with respect to $x$ is $e^x$. So for $y = 4e^t$, the derivative is $4e^t$.

Step4: Combine the derivatives

By the sum - difference rule of derivatives $(u\pm v\pm w)'=u'\pm v'\pm w'$, where $u = 6t^4$, $v=-4t$, and $w = 4e^t$. So $f'(t)=24t^3-4 + 4e^t$.

Answer:

$24t^3-4 + 4e^t$