Question

current objective
find the derivative of a sine or cosine function
question
find $f(x)$ where $f(x)= - 7xcos(x)$.
provide your answer below:
$f(x)=square$

Explanation:

Step1: Recall product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u=-7x$ and $v = \cos(x)$.

Step2: Find $u'$

Differentiate $u=-7x$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax)=a$ (where $a=-7$), we get $u'=-7$.

Step3: Find $v'$

Differentiate $v = \cos(x)$ with respect to $x$. The derivative of $\cos(x)$ is $-\sin(x)$, so $v'=-\sin(x)$.

Step4: Apply product - rule

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula $y'=u'v+uv'$.
We have $f'(x)=u'v + uv'=(-7)\cos(x)+(-7x)(-\sin(x))$.
Simplify the expression: $f'(x)=-7\cos(x)+7x\sin(x)$.

Answer:

$-7\cos(x)+7x\sin(x)$