#### To determine

**To state:**

Whether the given statement,"If f is differentiable, then ddxf(x)=f'(x)2f(x)" is true or false.

#### Answer

The given statement is true.

#### Explanation

The chain rule states that if g is differentiable at x and f is differentiable at gx, then the composite function Fx=f(gx) is differentiable at x and F'x is given by the product

F'x=f'gxg'(x)

Consider,

Fx= f(x)

let fu=u and gx=f(x)

Now, differentiating fu with respect to u

f'(u)=du12du

By power function rule,

d(xn)dx=nxn-1

⇒f'(u)=12u-12

⇒f'(u)=12u

Now, differentiating gx with respect to x,

gx=f(x)

⇒g'x=f'(x)

Now by the chain rule,

F'x=f'gxg'(x)

⇒F'x=f'f(x)f'(x)

⇒F'x=12f(x)f'(x)

⇒F'x=f'(x)2f(x)

⇒ddxf(x)=f'(x)2f(x)

Hence the statement is true.

**Final statement:**

The given statement is true.