Use differentials to estimate the value of $\sqrt[4]{\mathbf{1 . 4}}$. Compare the answer to the exact value of $\sqrt[4]{1.4}$ Round your answers to six decimal places, if required. You can use a calculator, spreadsheet, browser, etc. to calculate the exact value.

Step 1 ：Let's use the formula for differentials to estimate the value of \(\sqrt[4]{1.4}\). The formula for differentials is \(dy = f'(x) \cdot dx\), where \(f'(x)\) is the derivative of the function \(f(x)\) and \(dx\) is a small change in \(x\).

Step 2 ：In this case, our function is \(f(x) = \sqrt[4]{x}\), and we want to find the value of \(f(1.4)\). We can approximate this by finding the value of \(f(1)\) (which we know exactly) and then adding the differential \(dy\) for a small change \(dx = 0.4\).

Step 3 ：First, we need to find the derivative \(f'(x)\) of the function \(f(x) = \sqrt[4]{x}\).

Step 4 ：Then, we can calculate \(dy\) using the formula for differentials, and add this to \(f(1)\) to get our estimate for \(f(1.4)\).

Step 5 ：Finally, we can compare this estimate to the exact value of \(\sqrt[4]{1.4}\), which we can calculate using a calculator or other tool.

Step 6 ：The estimated value of \(\sqrt[4]{1.4}\) using differentials is approximately 1.1, and the exact value is approximately \(\boxed{1.087757}\).