Question

Find the derivative of the algebraic function. $$ h(x)=\left(x^{2}+1\right)^{2} $$

Short answer

The derivative of the function \( h(x)=(x^{2}+1)^{2} \) is \( h'(x) = 4x(x^2 + 1) \).

Step-by-step solution

Identify outer and inner functions

Here, the outer function is \( (u)^2 \) and the inner function is \( u=x^2+1 \).

Apply the chain rule

We use the chain rule which states that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function. The derivative of the outer function \( (u)^2 \) is \( 2(u) \) and the derivative of the inner function \( x^2+1 \) is \( 2x \). Therefore, the derivative of \( h(x) \) is given by \( h'(x) = 2(u) * (2x) \).

Substitute the inner function

Substitute \( u=x^2+1 \) back into the equation to get \( h'(x) = 2(x^2+1) * (2x) \).

Simplify the equation

This simplifies to \( h'(x) = 4x(x^2 + 1) \).