hw13 derivatives of trigonometric functions ($3.5) score: 3/7 answered: 3/7 question 4 if $f(x)=5x(sin x+cos…

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = 5x$ and $v=\sin x+\cos x$. First, find $u'$ and $v'$. The derivative of $u = 5x$ is $u'=5$. The derivative of $v=\sin x+\cos x$ is $v'=\cos x-\sin x$.

Step2: Calculate $f'(x)$

Using the product - rule $f'(x)=u'v + uv'$, we substitute $u = 5x$, $u' = 5$, $v=\sin x+\cos x$, and $v'=\cos x-\sin x$ into the formula. [ \begin{align*} f'(x)&=5(\sin x+\cos x)+5x(\cos x - \sin x)\ &=5\sin x+5\cos x + 5x\cos x-5x\sin x \end{align*} ]

Answer:

$5\sin x + 5\cos x+5x\cos x - 5x\sin x$