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Answer :
Final answer:
The derivative dy/dx for the function y = (6x^3 + 5)^(3/2) is found using the chain rule and power rule, resulting in 27x^2(6x^3 + 5)^(1/2), which matches option B.
Explanation:
The question asks to find the value of dy for the function y = (6x3 + 5)3/2. To find the derivative of y with respect to x, dy/dx, we use the chain rule and the power rule of differentiation. Let's first differentiate the outer function keeping the inner function the same, and then multiply by the derivative of the inner function.
Given y = (6x3 + 5)3/2, applying the chain rule, we get:
dy/dx = 3/2(6x3 + 5)1/2 × d/dx(6x3 + 5)
The differentiation of the inner function (6x3 + 5) concerning x is 18x2.
Therefore, dy/dx = 3/2(6x3 + 5)1/2 × 18x2 = 27x2(6x3 + 5)1/2.
So, the correct answer is option B) 9x2(6x3 + 5)1/2.
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