Problem 34 - University Calculus: Early Transcendentals

Question

Find the derivatives of the function. $$y=x^{-3 / 3}+\pi^{3 / 2}$$

Step-by-Step Solution

Answer

The derivative is \(-x^{-2}\).

Step 1: Simplify the Exponent

First, simplify the exponent in the term \( x^{-3/3} \). This simplifies to \( x^{-1} \) because the fraction \( -3/3 \) simplifies to -1.

Step 2: Use the Power Rule for Differentiation

The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Apply this rule to find the derivative of \( x^{-1} \). Differentiating, we get \( -1 \cdot x^{-1-1} = -x^{-2} \).

Step 3: Derive the Second Term

Identify that \( \pi^{3/2} \) is a constant term. The derivative of a constant is 0.

Step 4: Combine the Derivatives

Combine the derivatives you found in the previous steps. The derivative of the function \( y = x^{-3/3}+\pi^{3/2} \) is \( -x^{-2} + 0 = -x^{-2} \).

Key Concepts

Power Rule

The power rule is a fundamental principle in calculus, used to find the derivative of a function of the form \( x^n \). It simplifies the process of differentiation by establishing that the derivative of \( x^n \) is \( nx^{n-1} \). This rule is particularly useful because it streamlines the differentiation of polynomials.

• When applying the power rule, it's important to correctly identify the exponent \( n \).
• If the exponent is negative or a fraction, as in the expression \( x^{-3/3} \), it is advisable to simplify it first. For example, \( x^{-3/3} \) simplifies to \( x^{-1} \).

In practice, recognizing when to use the power rule involves:

• Identifying terms that can be expressed in the form of \( x^n \).
• Applying the rule iteratively to each applicable term in a polynomial or sum.

The power rule doesn't apply directly to all functions, but it's a key tool for polynomial expressions.

Derivative

In calculus, a derivative represents the rate at which a function is changing at any given point. It acts as a fundamental building block in understanding how functions behave. Differentiation, the process of finding a derivative, enables you to determine the slope of the tangent line to a graph at a particular point.

• For a function \( y = f(x) \), the derivative \( y' = f'(x) \) provides a mathematical way to express the rate of change.
• Not all functions change uniformly, hence calculative approaches like the power rule help ascertain these rates.

The derivative has several applications:

• It helps find velocity and acceleration in physics, as these are derivatives of position with respect to time.
• In economics, derivatives assist in calculating marginal costs and optimizing resources.
• They are pivotal in determining maxima and minima of functions, essential in various real-world optimizations.

Being adept at finding derivatives involves understanding and applying rules like the power rule and recognizing constant functions for ease of computation.

Constant Function

A constant function is one that remains the same regardless of the input value. Mathematically, a constant function is expressed as \( f(x) = c \), where \( c \) is a constant.

• The distinguishing feature of constant functions is their derivative, which is always zero.
• This happens because there is no change in the value of the function over its domain, hence the slope of the graph is zero.

In the problem given, \( \pi^{3/2} \) is identified as a constant.

• This term doesn’t change with \( x \), making it straightforward to differentiate.
• Applying this, the derivative of \( \pi^{3/2} \) is \( 0 \).

Constant functions simplify analysis in calculus by providing intuitive insights into equations, where straightforward terms do not affect the rate of change in a function. Understanding constant functions is important for quickly identifying parts of an expression that contribute zero to the derivative.