I’ve been trying to solve this derivative problem

Question

How can I find the derivative of this function using the quotient rule?

$$f(x)=\frac{x}{(x^4-2/x)^4}$$

Answer 1

HINT

Recall that

$$f(x)=\frac{g(x)}{h(x)} \implies f'(x)=\frac{g'(x)h(x)-h'(x)g(x)}{h^2(x)}$$

with

• $g(x)=x$
• $h(x)=(x^4-2/x)^4$

We van also simplify $f(x)$ to obtain

$$f(x)=\frac{x}{(x^4-2/x)^4}=\frac{x^5}{(x^5-2)^4}$$

with

• $g(x)=x^5$
• $h(x)=(x^5-2)^4$

Answer 2

$$f'(x)=\frac{1(x^4-2/x)^4-x\,4(4x^3+2/x^2)(x^4-2/x)^3}{(x^4-2/x)^8}.$$