Question

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{1}^{x} \sqrt[4]{t} d t $$

Short answer

The derivative of the function \(F(x)\) is \(F^{\prime}(x)=\sqrt[4]{x}\).

Step-by-step solution

- Application of the Second Fundamental Theorem of Calculus

According to the Second Fundamental Theorem of Calculus, if \(F(x)\) is defined as an integral from a constant (\(a\)) to \(x\) of a function (\(f(t)\)), i.e., \(F(x) = \int_a^x f(t) dt\), then the derivative of \(F(x)\) with respect to \(x\) is just the original function, i.e., \(F'(x) = f(x)\). Here, the function \(f(t)\) is \(\sqrt[4]{t}\), hence \(F^{\prime}(x)= \sqrt[4]{x}\).