If (y = 2(2x + sqrt{x})^{2}), find (frac{mathrm d y}{mathrm d x}).
-
A.
(2sqrt{x}(2x + sqrt{2})) -
B.
(4(2x + sqrt{x})(2 + frac{1}{2sqrt{x}})) -
C.
(4(2x + sqrt{x})(2 + sqrt{x})) -
D.
(8(2x + sqrt{x})(2 + sqrt{x}))
Correct Answer: Option B
Explanation
(y = 2(2x + sqrt{x})^{2})
Let (u = 2x + sqrt{x})
(y = 2u^{2})
(frac{mathrm d y}{mathrm d u} = 4u)
(frac{mathrm d u}{mathrm d x} = 2 + frac{1}{2sqrt{x}})
(therefore frac{mathrm d y}{mathrm d x} = (frac{mathrm d y}{mathrm d u})(frac{mathrm d u}{mathrm d x}))
= (4u(2 + frac{1}{2sqrt{x}}) )
= (4(2x + sqrt{x})(2 + frac{1}{2sqrt{x}}))