Problem 71 - Calculus for Biology and Medicine

Question

Find the derivative of $$ f(x)=\tan ^{3}\left(3 x^{3}-3\right) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 27x^2 \tan^2(3x^3 - 3) \sec^2(3x^3 - 3) \).

Step 1: Apply the Chain Rule

We start by recognizing that we need to use the Chain Rule to find the derivative of the given function. The function is composed of an outer function and an inner function. The outer function is the cube function, \( ( an(u))^3 \), and the inner function is \( an(3x^3 - 3) \). According to the Chain Rule, if we have a function \( y = g(h(x)) \), then \( \frac{dy}{dx} = g'(h(x)) \cdot h'(x) \).

Step 2: Differentiate the Outer Function

Differentiate the outer function, which is \( u^3 \) where \( u = \tan(3x^3 - 3) \). The derivative of \( u^3 \) with respect to \( u \) is \( 3u^2 \).

Step 3: Differentiate the Inner Function

Next, we differentiate the inner function \( \tan(v) \), where \( v = 3x^3 - 3 \). The derivative of \( \tan(v) \) with respect to \( v \) is \( \sec^2(v) \).

Step 4: Differentiate the Nested Inner Function

Lastly, differentiate the innermost function \( 3x^3 - 3 \). The derivative of \( 3x^3 - 3 \) with respect to \( x \) is \( 9x^2 \).

Step 5: Combine All Derivatives

Now, combine all our derivatives using the Chain Rule. First, multiply the outer derivative \( 3 \tan^2(3x^3 - 3) \) by the derivative of the inner function \( \sec^2(3x^3 - 3) \), and then multiply by the derivative of the innermost function \( 9x^2 \). This gives:\[ f'(x) = 3 \tan^2(3x^3 - 3) \cdot \sec^2(3x^3 - 3) \cdot 9x^2 \]

Step 6: Simplify the Expression

Simplify the derivative expression by multiplying the constants together and expressing it in a more concise form:\[ f'(x) = 27x^2 \tan^2(3x^3 - 3) \sec^2(3x^3 - 3)\]

Key Concepts

Chain Rule

The Chain Rule is a powerful tool in calculus used to differentiate composite functions. A composite function is simply a function within another function, like layers of an onion. When finding the derivative of such functions, the Chain Rule helps us navigate through these layers systematically.

• Imagine you have a function composed of two parts: an outer function, say, \(g(u)\), and an inner function, \(u = h(x)\). The Chain Rule allows us to differentiate this composition by multiplying the derivative of the outer function by the derivative of the inner function.
• In mathematical terms, if \(y = g(h(x))\), the derivative is determined as \(\frac{dy}{dx} = g'(h(x)) \cdot h'(x)\).
• This process resembles finding the rate of change of both layers, making it seamless to tackle more complex functions.

In our original exercise, the function \(f(x) = \tan^3(3x^3 - 3)\) includes different layers requiring us to delicately peel through them using the Chain Rule.

Trigonometric Functions

Trigonometric functions are fundamental in calculus, especially when applying differentiation. Common trigonometric functions include \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\), each having specific derivatives.

• The derivative of \(\tan(x)\) with respect to \(x\) is \(\sec^2(x)\), which plays a crucial role in our exercise.
• Understanding the behavior and differentiation of these trigonometric identities is essential, as they frequently appear in calculus problems.

In the provided function, the \(\tan\) function is nested within other functions. As we deploy the Chain Rule, recognizing and correctly differentiating \(\tan(3x^3 - 3)\) using the fact that its derivative is \(\sec^2\) is key to solving the problem accurately.
Trigonometric derivatives, like these, often serve pivotal steps that can make or break the successful differentiation of composite functions.

Differentiation Techniques

To successfully differentiate complex functions, mastery over various differentiation techniques is necessary. These techniques equip you with the ability to elegantly handle both simple and advanced problems.

• Begin with understanding basic derivatives, like power and exponential rules. For instance, the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\).
• Apply the Chain Rule when encountering composite functions, ensuring each component is systematically differentiated.
• Use product and quotient rules when functions are multiplied or divided, respectively.

In our specific exercise, the differentiation required breaking down the process:

• First, focus on differentiating the outermost layer, the cube function \(u^3\), using the power rule.
• Next, hone in on differentiating the trigonometric inner function \(\tan(v)\) using its specialized derivative \(\sec^2(v)\).
• Finally, tackle the innermost algebraic function \(3x^3 - 3\) using the power rule to conclude.

By combining these steps, resolved through systematic differentiation techniques, we find the desired derivative efficiently.