Problem 19 - University Calculus: Early Transcendentals

Question

Find \(d y\). $$y=x^{3}-3 \sqrt{x}$$

Step-by-Step Solution

Answer

The derivative, \(dy\), is \(3x^{2} - \frac{3}{2}x^{-1/2}\).

Step 1: Differentiate the Power Function

The function given is composed of two terms: \(x^3\) and \(-3\sqrt{x}\). Start by differentiating the first term \(x^3\). The derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\). For \(x^3\), this yields \(3x^{2}\).

Step 2: Rewrite the Square Root as a Power

The term \(-3\sqrt{x}\) can be rewritten using exponents: \(-3\sqrt{x} = -3x^{1/2}\). This is done to apply the power rule for derivatives.

Step 3: Differentiate the Square Root Function

Now differentiate the rewritten term \(-3x^{1/2}\). The derivative of \(x^{1/2}\) is \(\frac{1}{2}x^{-1/2}\). So the derivative of \(-3x^{1/2}\) is \(-3 \times \frac{1}{2}x^{-1/2} = -\frac{3}{2}x^{-1/2}\).

Step 4: Combine the Derivatives

Combine the derivatives of the two terms to find \(dy\). The derivative of \(x^3 - 3\sqrt{x}\) is \(3x^{2} - \frac{3}{2}x^{-1/2}\).

Key Concepts

Power Rule

Differentiation is a crucial concept in calculus, and one of the most frequently used rules for differentiating functions is the power rule. The power rule helps you find the derivative of a function with the form \(x^n\), where \(n\) is a constant exponent. The rule states that if \(y = x^n\), then the derivative \(\frac{dy}{dx} = nx^{n-1}\). This means multiplying the original exponent \(n\) with \(x\) raised to the power of \(n-1\).

Let's take the example of \(x^3\) from our exercise. Applying the power rule, we bring down the 3 in front of \(x\), and reduce the power by 1, resulting in the derivative \(3x^2\).

• The power rule simplifies differentiation of polynomial expressions.
• It is efficient for handling terms with positive and negative integer exponents.

Understanding the power rule is fundamental for tackling more complex differentiation problems.

Derivatives

Derivatives are central to calculus and represent the rate at which a function is changing at any point. They provide insights into the behavior of functions, such as velocity in physics, gradients in graphs, or the slope of a tangent line.

To find a derivative, especially using the power rule as shown, involves steps that calculate the rate of change of each function component. It involves rules like product, quotient, and chain rules, depending on the function's composition.

• Derivatives showcase how a function responds to changes in input.
• The act of differentiation is the process of finding a derivative.

In our example, differentiating \(y = x^3 - 3\sqrt{x}\) requires calculating the derivative of \(x^3\) and \(-3\sqrt{x}\) and then combining these to get \(3x^2 - \frac{3}{2}x^{-1/2}\).

Square Root Function

Functions involving square roots, like \(-3\sqrt{x}\), often require special consideration when differentiating. The square root can be rewritten as a power of \(x\) to apply the power rule. Specifically, \(\sqrt{x}\) is equivalent to \(x^{1/2}\).

Applying the power rule to \(-3x^{1/2}\), we can determine its derivative. By reducing the power by 1, we obtain \(x^{-1/2}\), and multiplying by the current coefficient and exponent gives \(-\frac{3}{2}x^{-1/2}\).

• Rewriting before applying the power rule makes differentiation easier.
• The negative exponent in the result reflects a division by the square root.

This approach transforms potentially difficult problems into straightforward applications of known rules.