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Difficulty Level:

Medium

MCQ

Angela tried to find the derivative of sin(x)x8\frac{\sin (x)}{x^{8}}. Here is her work:

Step 1: This is the quotient of sin(x)\sin ({x}) and x8x^{8}. So we should use the quotient rule.

Step 2:

ddx[sin(x)x8]=x8ddx[sin(x)]sin(x)ddx[x8](x8)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:

ddx[sin(x)]=cos(x)andddx[x8]=8x7\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:

ddx[sin(x)x8]=x8ddx[sin(x)]sin(x)ddx[x8](x8)2=x8cos(x)sin(x)8x7x16=xcos(x)8sin(x)x9\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?

Select all that apply :

Angel’s work is correct

Step 1 is incorrect. Angela should have used different rule and not the quotient rule.

Step 2 is incorrect. Angela did not state the correct quotient rule.

Step 3 is incorrect. Angela did not differentiate sin(x)\sin (x) correctly.

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