Question

Use the Chain Rule to find the derivative.
f(x) = (x5 − x3 − 1)2/9

Answer 1

To find the derivative of the function \( f(x) = (x^5 - x^3 - 1)^{2/9} \) using the Chain Rule, we will first identify the outer and inner functions.

Let:
- \( u = x^5 - x^3 - 1 \) (the inner function)
- \( f(x) = u^{2/9} \) (the outer function)

Now we apply the Chain Rule, which states:

\[
\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx}
\]

**Step 1: Compute \( \frac{df}{du} \)**

The derivative of \( u^{2/9} \) with respect to \( u \) is:

\[
\frac{df}{du} = \frac{2}{9} u^{\frac{2}{9} - 1} = \frac{2}{9} u^{-\frac{7}{9}}
\]

**Step 2: Compute \( \frac{du}{dx} \)**

Now we find the derivative of \( u = x^5 - x^3 - 1 \):

\[
\frac{du}{dx} = 5x^4 - 3x^2
\]

**Step 3: Combine using the Chain Rule**

Now we can substitute back into our formula using the Chain Rule:

\[
\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = \frac{2}{9} u^{-\frac{7}{9}} (5x^4 - 3x^2)
\]

Substituting \( u \) back in:

\[
\frac{df}{dx} = \frac{2}{9} (x^5 - x^3 - 1)^{-\frac{7}{9}} (5x^4 - 3x^2)
\]

**Final Answer:**

Thus, the derivative of \( f(x) \) is

\[
\frac{df}{dx} = \frac{2(5x^4 - 3x^2)}{9 (x^5 - x^3 - 1)^{\frac{7}{9}}}
\]