Problem 23 - Technical Mathematics with Calculus

Question

Find the derivative. $$D_{x}\left(4.8-7.2 x^{-2}\right)^{2}$$

Step-by-Step Solution

Answer

The derivative is \( D_{x}(4.8 - 7.2 x^{-2})^{2} = 138.24x^{-3} - 207.36x^{-5} \).

Step 1: Identify the Outer Function and Inner Function

First, identify the outer function and the inner function for the composition. Here, the outer function is a squaring function, f(u) = u^2, and the inner function is g(x) = 4.8 - 7.2x^{-2}.

Step 2: Apply the Chain Rule

Apply the chain rule which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. Symbolically: (f(g(x)))' = f'(g(x)) * g'(x).

Step 3: Compute the Derivative of the Outer Function

Compute the derivative of the outer function, using the power rule. For the squaring function, f(u) = u^2, the derivative is f'(u) = 2u.

Step 4: Compute the Derivative of the Inner Function

Compute the derivative of the inner function, g(x) = 4.8 - 7.2x^{-2}, using the power rule. The power rule states that the derivative of x^n is n*x^(n-1). Therefore, g'(x) = -7.2 * -2x^{-3} = 14.4x^{-3}.

Step 5: Apply Derivatives in the Chain Rule

Substitute the computed derivatives back into the chain rule formula. Multiply f'(g(x)) by g'(x) to get the derivative of the original function: D_x(4.8 - 7.2x^{-2})^2 = 2*(4.8 - 7.2x^{-2})*14.4x^{-3}.

Step 6: Simplify the Expression

Simplify the resulting expression by multiplying and combining like terms: D_x(4.8 - 7.2x^{-2})^2 = 2*(4.8 - 7.2x^{-2})*14.4x^{-3} = 2*4.8*14.4x^{-3} - 2*7.2*14.4x^{-5} = 138.24x^{-3} - 207.36x^{-5}.

Key Concepts

Composite Function Derivative

When dealing with complex functions, it's essential to understand how to differentiate using the chain rule. This rule is crucial when a function is composed of one function inside of another—the composite function. Let's break it down:

A composite function can be denoted as \( f(g(x)) \), where \( g(x) \) is the inner function and \( f(u) \) is the outer function, with \( u = g(x) \). The chain rule provides a method to find the derivative of this composite function. It states that the derivative of \( f(g(x)) \) is the derivative of \( f \) with respect to \( u \) (that is, \( f'(u) \)) evaluated at \( u = g(x) \) times the derivative of \( g \) with respect to \( x \) (\( g'(x) \) ). Symbolically, \[ (f(g(x)))' = f'(g(x)) \cdot g'(x) \].

Applying the chain rule to our sample exercise, the squaring function is the outer function and \( 4.8 - 7.2x^{-2} \) is the inner function. By differentiating both functions separately and then multiplying them, we obtain the derivative of the entire composite function.

Power Rule

The power rule is a fundamental technique used in calculus to differentiate functions that are monomials—the power functions of the form \( x^n \), where \( n \) is any real number. The rule is simple but powerful: To differentiate \( x^n \) with respect to \( x \), multiply \( n \) by \( x \) raised to the power of \( n-1 \). The rule can be expressed as \[ \frac{d}{dx}(x^n) = n \cdot x^{n-1} \].

In our example, the inner function includes a power of \( x \), which is \( -2 \). Applying the power rule, we take the exponent, multiply it by the coefficient (7.2), and then decrease the exponent by one to calculate the derivative. This results in \( 14.4x^{-3} \) for the derivative of the inner function.

Step by Step Derivative Calculation

A step-by-step approach to calculating derivatives ensures that complex problems are solved systematically. Starting with recognizing the types of functions involved (like identifying the outer and inner functions in a composite function), we proceed through the differentiation method appropriate for each function. When handling a composite function, the chain rule is employed, and the power rule is often a part of this process for differentiating polynomial expressions.

In the given exercise, for instance, after seeing that our function is composite, we first applied the power rule to the inner and outer functions, and then we used the chain rule to combine these derivatives. Each step builds on the previous one, and it's like constructing a mathematical puzzle where every piece has its place to form the complete picture of the derivative.

Simplifying Mathematical Expressions

The final step in the differentiation process often involves simplifying the mathematical expression to its most basic form. This step is key to making the result understandable and useful, particularly in applied contexts like physics and engineering.

After differentiating, we might end up with an expression that has complex coefficients or powers of \( x \). Multiplying through and combining like terms can significantly simplify these expressions. In our exercise, we multiply the derivatives of the inner and outer functions according to the chain rule, then combine like terms. The final expression \( 138.24x^{-3} - 207.36x^{-5} \) is simpler and more digestible than the original derived expression. Simplification allows us to easily see the relationship between the variable and its rate of change as represented by the derivative.