1. The problem asks to find the value of $x$ where the instantaneous rate of change of the function $f(x) = x^4$ is equal to 32. 2. The instantaneous rate of change of a function at a point is given by its derivative at that point. 3. First, find the derivative of $f(x) = x^4$ using the power rule: $$f'(x) = 4x^3$$ 4. Set the derivative equal to 32 to find the value of $x$: $$4x^3 = 32$$ 5. Divide both sides by 4: $$x^3 = 8$$ 6. Take the cube root of both sides: $$x = \sqrt[3]{8} = 2$$ 7. Therefore, the value of $x$ where the instantaneous rate of change of $f(x) = x^4$ is 32 is $x = 2$.