#### To determine

**To find:** The derivative of f(x)=1x2−13.

#### Answer

The derivative of f(x)=1x2−13 is f′(x)=−2x3(x2−1)x2−13.

#### Explanation

**Given:**

The function is f(x)=1x2−13.

**Derivative rule: The Power Rule combined with the Chain Rule**

If *n* is any real number and g(x) is differentiable function, then

ddx[g(x)]n=n[g(x)]n−1g′(x) (1)

**Calculation:**

The given function can be expressed as follows,

f(x)=1x2−13=1(x2−1)13f(x)=(x2−1)−13

Here. g(x)=x2−1.

Apply the Power Rule combined with the chain rule as shown in equation (1),

f′(x)=ddx(f(x))=ddx((x2−1)−13)=−13(x2−1)−13−1ddx(x2−1)=−13(x2−1)−43(ddx(x2)−ddx(1))

Simplify further and obtain the derivative of the function.

f′(x)=−13(x2−1)−43((2x2−1)+(0))=−13(x2−1)−43(2x)=−2x3(x2−1)(x2−1)13=−2x3(x2−1)(x2−1)3

Therefore, the derivative of f(x)=1x2−13 is f′(x)=−2x3(x2−1)x2−13.