Question

In Exercises \(19-26,(\) a) find \(d y,\) and (b) evaluate \(d y\) for the given value of \(x\) and \(d x\) . \(y=x^{3}-3 x, \quad x=2, \quad d x=0.05\)

Short answer

The value of dy when x is 2 and dx is 0.05 is 0.45.

Step-by-step solution

Determine the Derivative of the Function

The derivative of the function \(y = x^3 - 3x\) can be computed using the power rule, which states that the derivative of \(x^n\) is \(n x^{n-1}\). So, the derivative \(f'(x)\) equals \(3x^2 - 3\).

Find the General Differential dy

To find the differential dy, use the formula \(dy = f'(x) * dx\). Substituting \(f'(x) = 3x^2 - 3\) we have, \(dy = (3x^2 - 3)dx\).

Evaluate dy for Specific Values of x and dx

Now, substitute \(x = 2\) and \(dx = 0.05\) into the equation from step 2, we have \(dy = (3*(2)^2 - 3) * (0.05) = (12 - 3) * 0.05 = 0.45\).

Key concepts

Derivative of a Function

Understanding the derivative of a function is a foundational concept in differential calculus. It measures how a function's output value changes as its input value changes—in other words, the rate at which the function is changing at any point along its curve. Imagine traveling in a car; the derivative tells us not just the direction we're going, but how fast we're getting there.

For a given function, say, \( y = f(x) \), the derivative is often denoted by \( f'(x) \), \( \frac{dy}{dx} \) or \( \frac{df}{dx} \). It's like giving you a snapshot of the moment-by-moment action of the function \( f \) as \( x \) changes.

To find the derivative of a function like \( y = x^{3} - 3x \) from our exercise, we apply rules of differentiation to each term with respect to \( x \). This process allows us to predict the function's behavior and understand its properties in greater detail.

Power Rule Differentiation

One of the first and most useful rules in differentiation is the power rule. Simply put, if you have a function where the variable \( x \) is raised to a power \( n \)—expressed as \( x^n \)—the power rule tells us that the derivative of that function is \( n*x^{n-1} \) where \( n \) is a constant. It's like having a hammer in your toolbox; it's a basic, yet incredibly powerful tool.

Applying the Power Rule

Let's make this concrete with our exercise example, where we have \( y = x^{3} - 3x \). Using the power rule, we differentiate \( x^{3} \) to get \( 3x^{2} \) and \( -3x \) to get \( -3 \).

In general, the rule simplifies the process of finding derivatives, making it easier for students to move from understanding the theory behind derivatives to executing them in practical problems.

Evaluating Differentials

After determining the derivative of a function, the next step is often evaluating differentials. The differential \( dy \) or \( df \) can be thought of as an infinitesimally small change in the function's output \( y \) when its input \( x \) changes by an infinitesimally small amount \( dx \). It allows you to approximate changes in functions resulting from tiny nudges to their inputs.

For the function in our example, \( dy \) is calculated using the derivative \( f'(x) = 3x^{2} - 3 \) and the tiny input change \( dx \) as \( dy = f'(x) * dx \). If, as in our exercise, \( x = 2 \) and \( dx = 0.05 \) are given, we substitute these into the general differential to find the specific \( dy \), which gives us the approximate increase or decrease in \( y \) for the small increase in \( x \) by \( dx \). With this, you can now grasp how a function responds to small inputs, which is a crucial aspect of sensitivity analysis in many fields such as physics, economics, and engineering.