Question

find y for the following function. y = x^4 cos x y = □

Explanation:

Step1: Apply product - rule for first - derivative

The product - rule states that if $y = u\cdot v$, where $u = x^{4}$ and $v=\cos x$, then $y'=u'v + uv'$.
$u'=\frac{d}{dx}(x^{4}) = 4x^{3}$ and $v'=\frac{d}{dx}(\cos x)=-\sin x$.
So, $y'=4x^{3}\cos x - x^{4}\sin x$.

Step2: Apply product - rule for second - derivative

For $y' = 4x^{3}\cos x - x^{4}\sin x$, we find the derivative of each term separately.
For the first term $u_1 = 4x^{3}$ and $v_1=\cos x$, then $u_1' = 12x^{2}$ and $v_1'=-\sin x$. So the derivative of $4x^{3}\cos x$ is $12x^{2}\cos x-4x^{3}\sin x$.
For the second term $u_2=-x^{4}$ and $v_2 = \sin x$, then $u_2'=-4x^{3}$ and $v_2'=\cos x$. So the derivative of $-x^{4}\sin x$ is $-4x^{3}\sin x - x^{4}\cos x$.

Step3: Combine the results

$y''=(12x^{2}\cos x-4x^{3}\sin x)+(-4x^{3}\sin x - x^{4}\cos x)$.
$y''=(12x^{2}-x^{4})\cos x - 8x^{3}\sin x$.

Answer:

$(12x^{2}-x^{4})\cos x - 8x^{3}\sin x$