differentiate $f(x)=7xsin(8x)$. $f(x)=$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = 7x$ and $v=\sin(8x)$. First, find $u'$ and $v'$. The derivative of $u = 7x$ with respect to $x$ is $u'=7$.

Step2: Find derivative of $v$

To find the derivative of $v=\sin(8x)$, use the chain - rule. Let $t = 8x$, then $v=\sin(t)$. The derivative of $\sin(t)$ with respect to $t$ is $\cos(t)$, and the derivative of $t = 8x$ with respect to $x$ is 8. By the chain - rule, $v'=\cos(8x)\cdot8 = 8\cos(8x)$.

Step3: Apply product - rule formula

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula $y'=u'v + uv'$. We get $f'(x)=7\sin(8x)+7x\cdot8\cos(8x)$.

Step4: Simplify the expression

$f'(x)=7\sin(8x)+56x\cos(8x)$.

Answer:

$7\sin(8x)+56x\cos(8x)$