Problem 18 - Technical Mathematics with Calculus

Question

Derivative of In \(u\) Differentiate. $$y=5.06 \ln \sqrt{x^{2}-3.25 x}$$

Step-by-Step Solution

Answer

The derivative of the given function is \(y' = 2.53 \cdot \frac{1}{x^2 - 3.25x} \cdot (2x - 3.25)\).

Step 1: Rewrite the Function

Rewrite the natural logarithm of the square root as a power to simplify differentiation. Use the property \( \ln(x^n) = n \ln(x) \) to get \(y = 5.06 \ln((x^2 - 3.25x)^{1/2}) = 5.06 \cdot \frac{1}{2} \ln(x^2 - 3.25x)\).

Step 2: Apply the Constant Multiple Rule

Differentiate the function using the constant multiple rule, which states that the derivative of a constant times a function is the constant times the derivative of the function. In this case, apply the constant \(2.53 = 5.06 \cdot \frac{1}{2}\) to the result of the derivative of the logarithmic part.

Step 3: Differentiate the Logarithm

Differentiate the logarithmic function using the chain rule. The derivative of \( \ln(g(x)) \) is \(\frac{1}{g(x)}g'(x)\). Here \(g(x) = x^2 - 3.25x\), and its derivative \(g'(x)\) is \(2x - 3.25\).

Step 4: Chain Rule Application

Combine the results of the previous steps to find the derivative of the original function. The final derivative of the function is \(y' = 2.53 \cdot \frac{1}{x^2 - 3.25x} \cdot (2x - 3.25)\).

Key Concepts

Chain Rule

Understanding the chain rule is critical in calculus, especially when differentiating composite functions. In essence, it provides a method for finding the derivative of a function that is composed of other functions. The chain rule states that the derivative of a composite function is the derivative of the outside function evaluated at the inside function, multiplied by the derivative of the inside function.

Natural Logarithm Differentiation

When dealing with the natural logarithm function, represented by \(\ln(x)\), its differentiation is quite straightforward. The basic derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\), given that \(x\) is a positive real number. The differentiation becomes slightly more involved when the argument of the logarithm is itself a function of \(x\), as in the provided exercise. This is where the chain rule plays an essential role in finding the derivative effectively.

Constant Multiple Rule

The constant multiple rule in differentiation is a handy tool when a function is multiplied by a constant. The rule simply states that the derivative of a constant times a function is the constant multiplied by the derivative of the function. Symbolically, for any constant \(a\) and function \(f(x)\), the derivative \(\frac{d}{dx}(a \(f\))\) equals \(a \(\frac{d}{dx}f\)\). This property makes it more convenient to handle expressions with coefficients, as seen in our exercise with the coefficient \(5.06\), before applying the differentiation of the logarithmic function.

Mathematical Properties of Logarithms

Logarithms have several mathematical properties that make them easier to work with, especially when simplifying expressions before differentiation. One of these properties allows you to convert a logarithm of an exponentiated argument into a product, that is, \(\ln(x^n) = n \ln(x)\). This was used in our exercise to simplify the square root within the logarithm before proceeding with the differentiation. Applying the properties of logarithms to rewrite expressions can often simplify the differentiation process and lead to more manageable results.