Question

find the derivative of a sine or cosine function
question
find $f(x)$ where $f(x)=-\frac{6}{5}x^{4}cos(x)$.
provide your answer below:
$f(x)=\square$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u =-\frac{6}{5}x^{4}$ and $v=\cos(x)$.

Step2: Find the derivative of $u$

The derivative of $u =-\frac{6}{5}x^{4}$ using the power - rule $\frac{d}{dx}(ax^{n})=anx^{n - 1}$ is $u'=-\frac{6}{5}\times4x^{3}=-\frac{24}{5}x^{3}$.

Step3: Find the derivative of $v$

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

Step4: Calculate $f'(x)$

Using the product - rule $f'(x)=u'v+uv'$, we substitute $u, u', v, v'$:
$f'(x)=-\frac{24}{5}x^{3}\cos(x)-\frac{6}{5}x^{4}(-\sin(x))=-\frac{24}{5}x^{3}\cos(x)+\frac{6}{5}x^{4}\sin(x)$.

Answer:

$-\frac{24}{5}x^{3}\cos(x)+\frac{6}{5}x^{4}\sin(x)$