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?



<div id="displayOption0" class="markdown-body editormd-html-preview"><textarea>Angel's work is correct

</textarea><p>Angel’s work is correct</p>
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<div id="displayOption1" class="markdown-body editormd-html-preview"><textarea> Step 1 is incorrect. Angela should have used different rule and not the quotient rule.

</textarea><p> Step 1 is incorrect. Angela should have used different rule and not the quotient rule.</p>
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<div id="displayOption2" class="markdown-body editormd-html-preview"><textarea>Step 2 is incorrect. Angela did not state the correct quotient rule.

</textarea><p>Step 2 is incorrect. Angela did not state the correct quotient rule.</p>
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<div id="displayOption3" class="markdown-body editormd-html-preview"><textarea>Step 3 is incorrect. Angela did not differentiate $$\sin (x)$$ correctly.</textarea><p>Step 3 is incorrect. Angela did not differentiate <span class="editormd-tex"><span class="katex" style="font-size: 1.6em;"><span class="katex-mathml"><math><semantics><mrow><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\sin (x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height: 0.75em;"></span><span class="strut bottom" style="height: 1em; vertical-align: -0.25em;"></span><span class="base textstyle uncramped"><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span></span> correctly.</p>
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<div id="id_solution_display" class="markdown-body editormd-html-preview"><textarea>Angle Angela did every thing correctly,

</textarea><p>Angle Angela did every thing correctly,</p>
</div>