Problem 63 - Single Variable Calculus

Question

Find the derivative of the function. $$ y=2^{\cot x} $$

Step-by-Step Solution

Answer

The derivative of the function \(y = 2^{\cot x}\) is \(\frac{dy}{dx} = -\csc^2{x} \cdot \ln 2 \cdot 2^{\cot x}\).

Step 1: Take natural logarithm of both sides

Taking natural logarithm of both sides: \[\ln y = \ln\left(2^{\cot x}\right)\]

Step 2: Apply logarithmic properties

Using the power rule of logarithms, we get: \[\ln y = \cot x \cdot \ln 2\]

Step 3: Implicit differentiation

Now let's differentiate both sides with respect to x: \[\frac{d}{dx} (\ln y) = \frac{d}{dx} (\cot x \cdot \ln 2)\] By the chain rule, we have: \[\frac{1}{y} \cdot \frac{dy}{dx} = -\csc^2{x} \cdot \ln 2\]

Step 4: Isolate the derivative

Now, we'll isolate the derivative by multiplying both sides by \(y\): \[\frac{dy}{dx} = -\csc^2{x} \cdot \ln 2 \cdot y\] Since \(y=2^{\cot x}\), we can replace \(y\) with the given function, resulting in: \[\frac{dy}{dx} = -\csc^2{x} \cdot \ln 2 \cdot 2^{\cot x}\]

Step 5: Final Answer

The derivative of the function \(y = 2^{\cot x}\) is: \[\frac{dy}{dx} = -\csc^2{x} \cdot \ln 2 \cdot 2^{\cot x}\]

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a useful technique for finding the derivatives of functions that are challenging to differentiate directly. It is especially handy when dealing with exponential functions where the exponent is a variable expression. This method simplifies complex multiplication or division situations by transforming them into easier addition and subtraction forms using the properties of logarithms.

In the given problem, where we have a function like \(y = 2^{\cot x}\), we start by taking the natural logarithm of both sides. This simplifies the problem significantly. The power rule for logarithms states that \(\ln(a^b) = b \cdot \ln(a)\). Therefore, for our exercise, applying the natural logarithm leads to \(\ln y = \cot x \cdot \ln 2\).

This transformation allows for easier differentiation which brings us to the next steps involving the chain rule.

Chain Rule

The chain rule is a fundamental technique in calculus used when differentiating composite functions. It tells us how to differentiate a function based on its components.

Consider our previous step: \(\ln y = \cot x \cdot \ln 2\). To differentiate \(\ln y\) with respect to \(x\), we leverage the chain rule. Here, \(y\) is a function of \(x\), so differentiating \(\ln y\) with respect to \(x\) gives \(\frac{1}{y} \cdot \frac{dy}{dx}\).

For \(\cot x \cdot \ln 2\), notice that \(\ln 2\) is a constant multiplier, and we can focus on \(\cot x\). The derivative of \(\cot x\) is \(-\csc^2 x\), a crucial trigonometric derivative needed here. Thus, factoring in \(\ln 2\) results in a final differentiated form of \(-\csc^2 x \cdot \ln 2\) for this part.

Ultimately, the chain rule facilitates breaking down the problem into manageable parts relating these differentiated forms.

Trigonometric Functions

Trigonometric functions frequently appear in calculus problems, necessitating familiarity with their derivatives. When we see expressions like \(\cot x\) in our exercise, understanding its derivatives becomes essential.

Let's quickly revisit some basic derivatives of trigonometric functions:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).
• The derivative of \(\tan x\) is \(\sec^2 x\).
• The derivative of \(\cot x\) is \(-\csc^2 x\).
• The derivative of \(\csc x\) is \(-\csc x \cot x\).
• The derivative of \(\sec x\) is \(\sec x \tan x\).

For our problem, recognizing that the derivative of \(\cot x\) is \(-\csc^2 x\) is key to differentiating the function successfully. Understanding these derivatives will help you tackle any calculus problem involving trigonometric expressions.

Implicit Differentiation

Implicit differentiation is a technique used when a function cannot be easily solved for one variable. It works seamlessly when functions are expressed implicitly, or in forms where variables cannot be isolated on their own.

In our specific problem, beginning with \(\ln y = \cot x \cdot \ln 2\), we proceed by differentiating both sides with respect to \(x\). Keeping in mind that \(y\) is a function of \(x\), implicit differentiation is necessary. Applying it gives us \(\frac{1}{y} \cdot \frac{dy}{dx} = -\csc^2{x} \cdot \ln 2\).

This leads us to isolate \(\frac{dy}{dx}\), our derivative, by multiplying through by \(y\). Since \(y = 2^{\cot x}\), substituting back results in \(\frac{dy}{dx} = -\csc^2{x} \cdot \ln 2 \cdot 2^{\cot x}\).

Implicit differentiation elegantly handles situations where derivatives are nested or appear on both sides of an equation, completing our solution process.