Question

find the derivative of $y = 12x^{4}+\frac{5}{x^{7}} - 8\csc x$.\
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$\boldsymbol{y = 48x^{3}-\frac{35}{x^{8}} + 8\csc x\cot x}$\
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$\boldsymbol{y = 48x^{3}-\frac{35}{x^{8}} - 8\csc x\cot x}$\
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$\boldsymbol{y = 48x^{3}-\frac{35}{x^{6}} + 8\csc x\cot x}$\
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$\boldsymbol{y = 48x^{3}-\frac{5}{7x^{6}} + 8\csc x\cot x}$

Explanation:

Step1: Rewrite the rational term

Rewrite $\frac{5}{x^7}$ as $5x^{-7}$, so the function becomes:
$$y=12x^4+5x^{-7}-8\csc x$$

Step2: Differentiate the polynomial terms

Use power rule $\frac{d}{dx}(x^n)=nx^{n-1}$:

• Derivative of $12x^4$: $12\times4x^{4-1}=48x^3$
• Derivative of $5x^{-7}$: $5\times(-7)x^{-7-1}=-35x^{-8}=-\frac{35}{x^8}$

Step3: Differentiate the trigonometric term

Use derivative rule $\frac{d}{dx}(\csc x)=-\csc x\cot x$:
Derivative of $-8\csc x$: $-8\times(-\csc x\cot x)=8\csc x\cot x$

Step4: Combine all derivatives

Add the results from Steps 2 and 3:
$$y'=48x^3-\frac{35}{x^8}+8\csc x\cot x$$

Answer:

$y'=48x^3-\frac{35}{x^8}+8\csc x\cot x$ (the first option)