find the derivative of the function. y = e^4x cos(x) y =

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = e^{4x}$ and $v=\cos(x)$.

Step2: Find $u'$

Using the chain - rule, if $u = e^{4x}$, let $t = 4x$, then $\frac{du}{dt}=e^{t}$ and $\frac{dt}{dx}=4$. So, $u'=\frac{du}{dx}=\frac{du}{dt}\cdot\frac{dt}{dx}=4e^{4x}$.

Step3: Find $v'$

The derivative of $v=\cos(x)$ is $v'=-\sin(x)$.

Step4: Calculate $y'$

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula: $y'=u'v + uv'=4e^{4x}\cos(x)+e^{4x}(-\sin(x))=e^{4x}(4\cos(x)-\sin(x))$.

Answer:

$e^{4x}(4\cos(x)-\sin(x))$