Problem 62 - Thomas Calculus

Question

In Exercises \(55-68,\) use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$ y=\frac{1}{t(t+1)(t+2)} $$

Step-by-Step Solution

Answer

The derivative \( \frac{dy}{dt} \) is \(-\frac{1}{t^2(t+1)(t+2)} - \frac{1}{t(t+1)^2(t+2)} - \frac{1}{t(t+1)(t+2)^2}\)."

Step 1: Apply Natural Logarithm to Both Sides

Take the natural logarithm of both sides of the equation: \[\ln(y) = \ln\left(\frac{1}{t(t+1)(t+2)}\right)\]Using properties of logarithms, this becomes:\[\ln(y) = -\ln(t) - \ln(t+1) - \ln(t+2)\]

Step 2: Differentiate Both Sides with Respect to t

Differentiate the equation from Step 1 with respect to \(t\):\[\frac{d}{dt}[\ln(y)] = \frac{d}{dt}[-\ln(t) - \ln(t+1) - \ln(t+2)]\]This yields:\[\frac{1}{y}\frac{dy}{dt} = -\frac{1}{t} - \frac{1}{t+1} - \frac{1}{t+2}\]

Step 3: Solve for the Derivative \(\frac{dy}{dt}\)

Multiply both sides by \(y\) to isolate \(\frac{dy}{dt}\):\[\frac{dy}{dt} = y\left(-\frac{1}{t} - \frac{1}{t+1} - \frac{1}{t+2}\right)\]Substitute \(y = \frac{1}{t(t+1)(t+2)}\) back into the equation:\[\frac{dy}{dt} = \frac{1}{t(t+1)(t+2)}\left(-\frac{1}{t} - \frac{1}{t+1} - \frac{1}{t+2}\right)\]

Step 4: Simplify the Expression for \(\frac{dy}{dt}\)

Simplify the right-hand side:\[\frac{dy}{dt} = \frac{1}{t(t+1)(t+2)}\left(-\frac{1}{t} - \frac{1}{t+1} - \frac{1}{t+2}\right)\]The expression becomes:\[\frac{dy}{dt} = \frac{-1}{t^2(t+1)(t+2)} - \frac{1}{t(t+1)^2(t+2)} - \frac{1}{t(t+1)(t+2)^2}\]

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that represents how a function changes as its input changes. In simpler terms, if you have a function that describes a curve, the derivative of that function tells you the slope of the curve at any given point. In practical terms, understanding derivatives can help determine rates of change, such as speed, acceleration, and other physical phenomena.

When dealing with complex functions, especially those involving products and quotients, using logarithmic differentiation simplifies the process. This method is particularly useful when the function is a product or a quotient of several expressions, as it helps manage the cumbersome algebra. In this exercise, for the function- \(y = \frac{1}{t(t+1)(t+2)}\), using logarithmic differentiation simplifies finding the derivative significantly. By taking the natural logarithm of both sides and differentiating, we break down a complex expression into manageable parts.

Natural Logarithm

A natural logarithm is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm is denoted by \(\ln\). It is frequently used in mathematics due to its natural properties relating to the derivative and integral calculus. Its unique properties make it particularly useful in dealing with exponential growth and decay problems.

In this exercise, taking the natural logarithm of both sides - \(\ln(y) = \ln\left(\frac{1}{t(t+1)(t+2)}\right)\) allows us to transform a complex expression into a sum of simpler terms using the logarithmic property \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\). Applying these rules turns multiplicative relationships into additive ones, making it easier to differentiate. This transformation is a key step that simplifies finding the derivative.

Simplification

Simplification is the process of rewriting expressions in a more concise and convenient form. This can make it easier to understand, evaluate, and apply expressions, especially when performing calculus operations such as differentiation or integration.

When dealing with derivatives obtained through logarithmic differentiation, simplification helps us reach a more actionable form. For example, after using logarithmic differentiation, we are left with an expression:- \[\frac{dy}{dt} = \frac{1}{t(t+1)(t+2)}\left(-\frac{1}{t} - \frac{1}{t+1} - \frac{1}{t+2}\right)\]Simplifying involves expanding the expression to:- \[\frac{dy}{dt} = \frac{-1}{t^2(t+1)(t+2)} - \frac{1}{t(t+1)^2(t+2)} - \frac{1}{t(t+1)(t+2)^2}\]Each term here is simpler and easier to analyze, providing a clearer understanding of how the original function changes with respect to \(t\). This can aid in solving problems related to rates of change or optimization, which often appear in real-world applications.