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Answer :
Final answer:
The derivative of f(x) = cot(5x^7 - 1) is found using the chain rule. The correct derivative is -csc^2(5x^7 - 1) multiplied by the derivative of the inner function, which is 35x^6. Thus, the correct answer is option b) -5csc^2(5x^7 - 1) * 7x^6.
Explanation:
To differentiate the function f(x) = cot(5x⁷- 1), we use the chain rule and the fact that the derivative of cot(x) is -csc2(x). The chain rule tells us that we need to take the derivative of the outer function (in this case, cotangent), and then multiply it by the derivative of the inner function (5x⁷ - 1 in this case).
The derivative of cot(x) is given by:
d(cot(x))/dx = -csc2(x)
The derivative of 5x⁷ - 1 is:
d(5x⁷ - 1)/dx = 35x⁶
Thus, the derivative of f(x) is:
f'(x) = -csc2(5x⁷ - 1) \\times 35x⁶
So the correct answer is:
b) -5csc2(5x⁷ - 1) \\times 7x⁶
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