# Differential of a Function Calculator

## Solution

Find the second point: $$x_{0} + \Delta x_{0} = 1 + \frac{1}{4} = \frac{5}{4}$$$. Evaluate the function at the two points: $$f{\left(x_{0} + \Delta x_{0} \right)} = f{\left(\frac{5}{4} \right)} = \frac{125}{64}$$$, $$f{\left(x_{0} \right)} = f{\left(1 \right)} = 1$$$. According to the definition: $$\Delta y = f{\left(x_{0} + \Delta x_{0} \right)} - f{\left(x_{0} \right)} = \frac{125}{64} - 1 = \frac{61}{64}$$$.

Find the derivative: $$f^{\prime }\left(x\right) = 3 x^{2}$$$(the steps can be seen here). Evaluate the derivative at $$x_{0} = 1$$$: $$f^{\prime }\left(1\right) = 3$$$. The differential is defined as $$dy = f^{\prime }\left(x_{0}\right) \Delta x_{0} = \left(3\right)\cdot \left(\frac{1}{4}\right) = \frac{3}{4}$$$.

Note that the value of $$dy$$$becomes closer to $$\Delta y$$$ as $$\Delta x_0 \to 0$$$. ## Answer $$\Delta y = \frac{61}{64} = 0.953125$$$A, $$dy = \frac{3}{4} = 0.75$$\$A.

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