(f(x) = (x^{2} + 3)^{2}) is defines on the set of real numbers, R. Find the gradient of f(x) at x = (frac{1}{2}).
-
A.
4.0 -
B.
6.5 -
C.
5.0 -
D.
10.6
Correct Answer: Option B
Explanation
(f(x) = (x^{2} + 3)^{2})
Using the chain rule, (frac{mathrm d y}{mathrm d x} = frac{mathrm d y}{mathrm d u} times frac{mathrm d u}{mathrm d x})
Let (u = x^{2} + 3) so that (y = u^{2})
(frac{mathrm d y}{mathrm d u} = 2u)
(frac{mathrm d u}{mathrm d x} = 2x)
(therefore frac{mathrm d y}{mathrm d x} = 2u(2x) = 4xu)
But (u = x^{2} + 3),
(frac{mathrm d y}{mathrm d x} = 4x(x^{2} + 3))
At (x = frac{1}{2}, frac{mathrm d y}{mathrm d x} = 4(frac{1}{2})((frac{1}{2})^{2} + 3))
= (2 times frac{13}{4} = frac{13}{2} = 6.5)