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f(x)=(x2+3)2 is defines on the set of real num...

f(x)=(x2+3)2 is defines on the set of real numbers, R. Find the gradient of f(x) at x = 12

A.

4.0

B.

6.5

C.

5.0

D.

10.6

Correct answer is B

f(x)=(x2+3)2

Using the chain rule, dydx=dydu×dudx

Let u=x2+3 so that y=u2

dydu=2u

dudx=2x

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