Question

Find the derivatives of the following functions using the chain rule.

Answer 1

To find:

The derivative of a function:

$f(x)=(x^4+1)^{-2}$

Solution:

It is known that the chain rule of differentiation is as follows:

$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$

So, the differentiation of the given function is as follows:

$\begin{gathered} f^{\prime}(x)=((x^4+1)^{-2})^{\prime} \\ =-2(x^4+1)^{-3}(x^4+1)^{\prime} \\ =-2(x^4+1)^{-3}(4x^3) \\ =-8x^3(x^4+1)^{-3} \end{gathered}$

Thus, the answer is:

$f^{\prime}(x)=-8x^3(x^4+1)^{-3}$