Question

In Exercises \(31-42,\) find \(d y / d x\). $$y=\sqrt[4]{x}$$

Short answer

The derivative of \(y=\sqrt[4]{x}\) or \(y=x^{1/4}\) is \((1/4)/\sqrt[4]{x^3}\) or \((1/4)x^{-3/4}\).

Step-by-step solution

Rewrite the function

Rewrite the equation with fractional exponents instead of a roots. This is important as it makes the function easier to differentiate. So, rewrite the function as \(y = x^{1/4}\).

Apply the power rule

The power rule states that the derivative of x to the power of n, \(x^n\), is \(nx^{n-1}\). Appy the rule to the function, which gives \(dy/dx = (1/4)x^{-3/4}\). This is the derivative of the given function.

Simplify the function

The derivative function can be simplified further by rewriting \(x^{-3/4}\) as \(1/x^{3/4}\) or \(1/\sqrt[4]{x^3}\). The derivative, dy/dx, simplifies to \((1/4)/\sqrt[4]{x^3}\).