Angela tried to find the derivative of $\frac{\sin (x)}{x^{8}}$. Here is her work:

Step 1: This is the quotient of $\sin ({x})$ and $x^{8}$. So we should use the quotient rule.

Step 2:

$\frac{d}{d x}[\frac{\sin (x)}{x^{8}}]=\frac{x^{8} \frac{d}{d x}[\sin (x)]-\sin (x) \frac{d}{d x}[x^{8}]}{(x^{8})^{2}}$

Step 3: Finding the derivatives of the factors:

$\frac{d}{d x}[\sin (x)]=\cos (x) \quad and \quad \frac{d}{d x}[x^{8}]=8 x^{7}$

Step 4: Putting it all together:

$\frac{d}{d x}[\frac{\sin (x)}{x^{8}}]=\frac{x^{8} \frac{d}{d x}[\sin (x)]-\sin (x) \frac{d}{d x}[x^{8}]}{(x^{8})^{2}}=\frac{x^{8} \cos (x)-\sin (x) 8 x^{7}}{x^{16}}=\frac{x \cos (x)-8 \sin (x)}{x^{9}}$

Is Ângela’s work correct? If not, what’s her mistake?