Problem 16 - Calculus: Early Transcendental Functions

Question

Evaluate the derivative using properties of logarithms where needed. $$\frac{d}{d x}(\ln \sqrt{\frac{x^{3}}{x^{5}+1}})$$

Step-by-Step Solution

Answer

The derivative of the function is \(\frac{1}{2}(\frac{3}{x} - \frac{5x^{4}}{x^{5} + 1})\).

Step 1: Re-write using properties of logarithms

The function can be re-written using properties of logarithms. Logarithm of a square root can be written as one-half of the logarithm. The quotient inside the square root can be split into subtraction of the logarithms. So, the given function becomes \(\frac{1}{2}(\ln x^{3} - \ln (x^{5}+1))\).

Step 2: Apply logarithmic differentiation

Applying logarithmic differentiation, where the derivative of \(\ln u\) is \(\frac{u'}{u}\), the differentiated function becomes: \(\frac{1}{2}(3x^{2}/x^{3} - 5x^{4}/(x^{5} + 1))\).

Step 3: Simplify the expression

Simplify the expression by factoring out common terms and reducing complex fractions. The simplified form is \(\frac{1}{2}(\frac{3}{x} - \frac{5x^{4}}{x^{5} + 1})\) . This is the derivative of the given function.

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique for finding the derivatives of complex functions. It is particularly useful when dealing with functions that are products, quotients, or powers. This technique involves taking the natural logarithm of both sides of an equation and then differentiating.

The advantage of logarithmic differentiation is that it simplifies the differentiation process by transforming complicated multiplication or division inside a function into an addition or subtraction outside, thanks to the properties of logarithms. Instead of directly differentiating the complex function, you simplify it first using logarithms and then differentiate. This is especially helpful with functions that involve a lot of variables and operations.

In our original problem, logarithmic differentiation allows us to handle the complex quotient under the square root before differentiation occurs, thus easing the calculation process.

Properties of Logarithms

The properties of logarithms are essential tools used in simplifying complex expressions. These properties include:

• The Product Rule: \(\ln(ab) = \ln a + \ln b\).


• The Quotient Rule: \(\ln(a/b) = \ln a - \ln b\).


• The Power Rule: \(\ln(a^b) = b\ln a\).


• The Logarithm of a Root: \(\ln(\sqrt{a}) = \frac{1}{2}\ln a\).

These rules help break down a complex logarithmic expression into simpler parts that are easier to differentiate.

In the original exercise, the function \(\ln \sqrt{\frac{x^3}{x^5+1}}\) is re-written by first using the logarithm of a root property and then applying the quotient rule. This provides a more straightforward expression to differentiate, \(\frac{1}{2}(\ln x^3 - \ln (x^5+1))\), which is manageable and clear.

Simplifying Expressions

Simplifying expressions is often the final step in calculus problems. It involves manipulating the expression into its simplest form. This is where properties such as factoring, canceling common terms, and reducing fractions play a crucial role.

After applying logarithmic differentiation, the goal is often to make the resulting expression simpler. This can involve factors like breaking down complex fractions into more basic terms, reducing terms with common factors, or adjusting coefficients to their simplest form.

In our exercise, once we applied the differentiation, we moved from a more complex differentiated expression to \(\frac{1}{2}(\frac{3}{x} - \frac{5x^4}{x^5 + 1})\). Through simplifying, it is easier to interpret and use the final answer for further calculations.