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Answer :
Final answer:
The derivative is C) 3x^(-5/2) + 2x^(-1/2) + 7x^6 - 2(dy/dx).
Explanation: To find the derivative of y = 3x^(-3/2) + 2x^(-1/2) + x^7 - 2y, we need to apply the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1). For the term -2y, we need to apply the chain rule, which states that the derivative of y with respect to x is (dy/dx) times the derivative of y with respect to its own variable.
Applying the power rule to the first three terms, we get:
d(3x^(-3/2))/dx = 3 (-3/2) x^((-3/2)-1) = -9x^(-5/2) d(2x^(-1/2))/dx = 2 (-1/2) x^((-1/2)-1) = -x^(-3/2) d(x^7)/dx = 7 * x^(7-1) = 7x^6
Now, applying the chain rule to the term -2y, we get:
d(-2y)/dx = -2 * (dy/dx)
Combining all these terms, we get:
dy/dx = -9x^(-5/2) + -x^(-3/2) + 7x^6 - 2(dy/dx)
To solve for dy/dx, we can add 2(dy/dx) to both sides and then divide by 3:
3(dy/dx) = -9x^(-5/2) + -x^(-3/2) + 7x^6 dy/dx = (-9x^(-5/2) + -x^(-3/2) + 7x^6)/3 = -9x^(-5/2)/3 - x^(-3/2)/3 + 7x^6/3 = 3x^(-5/2) + 1/3*x^(-3/2) + 7x^6/3 - 2(dy/dx)
Since we want to express dy/dx in terms of x only, we can substitute dy/dx on the right side with a new variable v:
v = 3x^(-5/2) + 1/3x^(-3/2) + 7x^6/3 -> dv/dx = d²y/(dxx), which is what we want. Therefore, our final answer is C) 3x^(-5/2) + 1/3x^(-3/2) + 7x^6/3.
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