Problem 58 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. $$ f(x)=\log \left(\sqrt[3]{\tan x^{2}}\right) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{2x}{3} \cdot \frac{\sec^2(x^2)}{\tan x^2} \).

Step 1: Rewrite the Function

The given function is \( f(x) = \log \left( \sqrt[3]{\tan x^2} \right) \). Begin by rewriting the expression inside the logarithm using properties of exponents: \( \sqrt[3]{\tan x^2} = (\tan x^2)^{1/3} \). Thus, the function becomes \( f(x) = \log \left( (\tan x^2)^{1/3} \right) \).

Step 2: Use Logarithm Properties

Apply the logarithmic identity: \( \log(a^b) = b \cdot \log(a) \) to the expression \( \log \left( (\tan x^2)^{1/3} \right) \). This gives us \( f(x) = \frac{1}{3} \log (\tan x^2) \).

Step 3: Differentiate Using the Chain Rule

Differentiate \( f(x) = \frac{1}{3} \log (\tan x^2) \) with respect to \( x \). First, apply the chain rule to the logarithmic term: \( \frac{d}{dx} [ \log(u)] = \frac{1}{u} \cdot \frac{du}{dx} \) where \( u = \tan x^2 \). We need \( \frac{d}{dx}(\tan x^2) \).

Step 4: Differentiate \( \tan x^2 \) Using the Chain Rule

Let \( u = x^2 \), so \( \tan x^2 = \tan(u) \). The derivative of \( \tan(u) \) with respect to \( u \) is \( \sec^2(u) \). Thus, \( \frac{d}{du}(\tan x^2) = \sec^2(x^2) \cdot \frac{du}{dx} = \sec^2(x^2) \cdot 2x \).

Step 5: Substitute Back Into the Derivative

Using steps 3 and 4, substitute the derivative of \( \tan x^2 \), we get: \( \frac{d}{dx} [ \log(\tan x^2)] = \frac{1}{\tan x^2} \cdot \sec^2(x^2) \cdot 2x = \frac{2x \sec^2(x^2)}{\tan x^2} \).

Step 6: Simplify the Expression

Multiply this result by \( \frac{1}{3} \), as per step 2: \( \frac{1}{3} \cdot \frac{2x \sec^2(x^2)}{\tan x^2} = \frac{2x}{3} \cdot \frac{\sec^2(x^2)}{\tan x^2} \). This is the derivative of the original function.

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus that is used when dealing with composite functions. When one function is nested inside another, like in the case of a function of the form \( f(g(x)) \), the chain rule helps us differentiate it. The essence of the chain rule is to differentiate the outer function and multiply it by the derivative of the inner function.

Mathematically, if \( y = f(u) \) and \( u = g(x) \), then the derivative of \( y \) with respect to \( x \) is given by:

• \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \)

In the given exercise, to differentiate \( \log (\tan x^2) \) with respect to \( x \), we first differentiate \( \log u \) with respect to \( u \) and then multiply by the derivative of \( u = \tan x^2 \) with respect to \( x \). This step involves using both the chain rule and trigonometric differentiation to successfully break down the components into manageable parts.

Logarithm Properties

Logarithm properties transform complex expressions into simpler forms, making differentiation much easier. The most commonly used logarithm property in calculus is \( \log(a^b) = b \cdot \log(a) \). This allows us to bring down exponents as coefficients, streamlining the differentiation process.

In the exercise, by rewriting \( \log \left( (\tan x^2)^{1/3} \right) \) as \( \frac{1}{3} \log(\tan x^2) \), we utilized this property, effectively turning a power into a simpler multiplication problem. This step significantly simplifies the differentiation, as the exponent is now a constant factor that can be readily incorporated into the derivative.

Similarly, understanding logarithms is helpful for converting difficult multiplication or division problems into simpler addition or subtraction problems using properties like:

• \( \log(ab) = \log a + \log b \)
• \( \log \left(\frac{a}{b}\right) = \log a - \log b \)

These transformations are invaluable for algebraically preparing functions for differentiation.

Trigonometric Differentiation

Trigonometric differentiation involves finding the derivative of functions that have trigonometric components such as sine, cosine, and tangent. Essential derivatives to remember include:

• \( \frac{d}{dx} [ \sin(x) ] = \cos(x) \)
• \( \frac{d}{dx} [ \cos(x) ] = -\sin(x) \)
• \( \frac{d}{dx} [ \tan(x) ] = \sec^2(x) \)

In the exercise, we dealt specifically with \( \tan x^2 \). The derivative of \( \tan(u) \) is \( \sec^2(u) \), and by substituting \( u = x^2 \), you apply the chain rule, yielding \( \sec^2(x^2) \cdot \frac{d}{dx}(x^2) = \sec^2(x^2) \cdot 2x \).

This application of the trigonometric differentiation formulas alongside the chain rule allows us to deconstruct complex trigonometric expressions, facilitating their differentiation.

Exponent Properties

Exponent properties simplify expressions, which is crucial when preparing functions for differentiation. One key property is rewriting radicals and roots as fractional exponents. For example, \( \sqrt[3]{a} \) can be expressed as \( a^{1/3} \). This transforms roots into simple multiplication problems which are more manageable.

In the problem provided, \( \sqrt[3]{\tan x^2} \) is rewritten as \( (\tan x^2)^{1/3} \). By doing so, it helps ease the task of applying logarithm properties and facilitates differentiation. Exponent rules paired with logarithm properties are especially powerful.

A few important exponent rules include:

• \( (a^m)^n = a^{m\cdot n} \)
• \( a^0 = 1 \)
• \( a^{-n} = \frac{1}{a^n} \)

Utilizing these properties provides a structured method to prep functions for differentiation, especially when they include nested functions or compounded operations.