High School

Find the derivative of \( y = 12x^4 + \left(\frac{5}{x^7}\right) - 8\csc x \).

A. \( y' = 48x^3 - \left(\frac{35}{x^8}\right) + 8\csc x \cot x \)

B. \( y' = 48x^3 - \left(\frac{35}{x^8}\right) - 8\csc x \cot x \)

C. \( y' = 48x^3 - \left(\frac{5}{7x^6}\right) + 8\csc x \cot x \)

D. \( y' = 48x^3 - 48x^3 + 8\csc x \cot x \)

Answer :

The derivative of the function y = 12x⁴ + (5/x⁷) - 8cscx is y' = 48x³ - (35/x⁸) + 8cscxcotx, which matches option A.

To find the derivative of the function y = 12x⁴ + (5/x⁷) - 8cscx, we'll use basic differentiation rules including the power rule, the quotient rule, and the derivative of trigonometric functions.

Step 1: Differentiating 12x⁴ using the power rule gives us 48x³.

Step 2: For the term (5/x⁷), which is 5x-7, the power rule yields -(35/x⁸).

Step 3: The derivative of -8cscx is 8cscxcotx, as the derivative of csc(x) is -csc(x)cot(x) and we have a negative sign initially.

Combining all the derivatives, we obtain y' = 48x³ - (35/x⁸) + 8cscxcotx, which corresponds to option A.

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