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In differential calculus, finding the derivative of a function is like unlocking its secret code. The derivative represents how a function changes as its input changes. To calculate this, we apply the rules of differentiation to our function. In the exercise, we're given the function \[ f(x) = x^3 - x \] which means we need to differentiate it with respect to \(x\). To do so, we use calculus rules like the power rule, which states that for any term \(x^n\), the derivative is \(nx^{n-1}\). This makes it easy to handle polynomials:

• For the term \(x^3\), the derivative is \(3x^2\).
• For \(-x\), it's simply \(-1\) (since the derivative of \(x\) is 1).

Putting it all together, the derivative of the function is \[ f'(x) = 3x^2 - 1 \] Understanding how to calculate a derivative can help you predict how functions behave, which is especially useful in physics, engineering, or any field where rates of change are important.

Once you have the derivative, the next step is to compute differentials. A differential gives you an approximation of how much a function changes, when you make a small change in the input. If you know the derivative at a specific point, you can estimate the function's change around that point using the formula:\[ df(a) = f'(a) \, dx \]In the exercise, we have already calculated that the derivative at \( a = 1 \) is \( f'(1) = 2 \). Additionally, we're given a small change \( dx = 0.1 \). By substituting these into our differential formula, we find:

• The approximate change in function is \( df(1) = 2 \times 0.1 = 0.2 \).

This computation of differentials is a simple yet powerful tool as it aids in making predictions about the behavior of functions in response to small perturbations in their inputs. It's often used in engineering and science for practical calculations.

Calculus is a field teeming with rules that help us maneuver through the fascinating world of change and motion. In our derivative calculation process, we employ some basic calculus rules:

• Power Rule: This rule lets you differentiate terms of the form \(x^n\) easily by making them \(nx^{n-1}\).
• Sum Rule: When you differentiate a sum of two functions, you can simply differentiate each function separately and add them together.
• Constant Rule: If a function is a constant, i.e., it's not dependent on \(x\), its derivative is zero.

Applying these rules simplifies the process of finding derivatives and differentials. Once you master them, you can tackle more complex functions with assurance. These rules form the backbone of many problem-solving techniques in differential calculus, making them vital tools for students and professionals alike.