Question

For the following exercises, find the derivative \(d y / d x\). (You can use a calculator to plot the function and the derivative to confirm that it is correct.) $$ \text { [T] } y=\ln (\tan (3 x)) $$

Short answer

\( \frac{dy}{dx} = \frac{3}{\sin(3x)\cos(3x)} \)

Step-by-step solution

Recognize the Form of the Composition

We are given the function \( y = \ln(\tan(3x)) \). This function is a composition of three functions: the natural logarithm \( \ln(u) \), the tangent \( \tan(v) \), and the linear function \( 3x \). We will differentiate using the chain rule.

Apply the Chain Rule for Differentiation

The chain rule states that if a function is a composition like \( y = f(g(h(x))) \), then the derivative is \( \frac{dy}{dx} = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \). Identify each part: \( u = \tan(3x) \) and \( v = 3x \). Start differentiating from the outermost function.

Differentiate the Outer Function (Natural Logarithm)

The derivative of the natural logarithm \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). So, for \( y = \ln(\tan(3x)) \), the derivative with respect to \( \tan(3x) \) is \( \frac{1}{\tan(3x)} \).

Differentiate the Intermediate Function (Tangent)

The derivative of \( \tan(v) \) with respect to \( v \) is \( \sec^2(v) \). Here, \( v = 3x \), so the derivative with respect to \( 3x \) is \( \sec^2(3x) \).

Differentiate the Innermost Function (Linear Function)

The derivative of \( 3x \) with respect to \( x \) is simply \( 3 \).

Combine Derivatives Using the Chain Rule

Combine all the derivatives together using the chain rule:\[ \frac{dy}{dx} = \frac{1}{\tan(3x)} \times \sec^2(3x) \times 3. \] This simplifies to:\[ \frac{dy}{dx} = \frac{3\sec^2(3x)}{\tan(3x)}. \]

Simplify the Derivative Expression

Recall that \( \sec^2(3x) = \frac{1}{\cos^2(3x)} \) and \( \tan(3x) = \frac{\sin(3x)}{\cos(3x)} \). Substitute these back to further simplify:\[ \frac{dy}{dx} = 3 \times \frac{1}{\cos^2(3x)} \times \frac{\cos(3x)}{\sin(3x)} = \frac{3}{\sin(3x)\cos(3x)}. \]

Key concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus that is essential when dealing with compositions of functions. Imagine you have a function nested inside another, like a Russian Matryoshka doll. To differentiate such a composition, the Chain Rule breaks it down into simpler parts.

Consider you have a composed function like \( f(g(h(x))) \). The Chain Rule guides us to differentiate it by taking the derivative of the outer function evaluated at the inner function, and multiply it by the derivatives of the inner functions step by step. This can be mathematically expressed as:

\[ \frac{dy}{dx} = f'(g(h(x))) imes g'(h(x)) imes h'(x) \]

Let's apply it to a practical example. Given \( y = \ln(\tan(3x)) \), observe how the function \(\ln(u))\), with \(u = \tan(v))\) and \(v = 3x \), breaks into these neatly differentiated pieces. Start from the outer function and work your way in, multiplying as you go. This layered differentiation ensures each piece of the function is accounted for.

Logarithmic Functions

Logarithmic functions, such as the natural logarithm \(\ln(x)\), play a crucial role in calculus. They are particularly handy for transforming multiplicative relationships into additive ones. This makes solving equations and plotting graphs more manageable.

The derivative of a logarithmic function is straightforward: for \( y = \ln(u) \), the derivative is \( \frac{1}{u} \) multiplied by the derivative of \(u\) itself. This accounts for changes in \(u\) as \(y\) changes. In our example, the function \(y = \ln(\tan(3x))\) involves the logarithm of a trigonometric function, \( \tan(3x) \).

Understanding logarithmic differentiation helps simplify complex relationships and solve growth-related problems in both pure and applied mathematics. It converts exponents and other tricky mathematical operations into a more digestible form, suitable for in-depth analysis.

Trigonometric Functions

Trigonometric functions, like sine, cosine, and tangent, are the foundation of understanding waves, oscillations, and cycles. In calculus, differentiating these functions is critical for analyzing how curves behave.

For our given problem, we focus on the tangent function \(\tan(v)\), where \(v = 3x\). The derivative of \(\tan(v)\) is \(\sec^2(v)\), which plays a crucial role in the solution. With \(v = 3x\), this becomes \(\sec^2(3x)\), illustrating how trigonometric derivatives combine with the chain rule.

Knowing how to work with these derivatives allows for predicting the behavior of periodic phenomena. This includes everything from sound waves to alternating current electricity. In calculus, these derivatives are stepping stones to understanding these complex patterns and forms.