Problem 20 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\tan ^{4}\left(x^{3}\right) $$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 12x^2 \tan^3(x^3) \sec^2(x^3)\).

Step 1: Identify the Structure of the Function

The given function is composed of two main parts: an outer function, which is raising to the fourth power, and an inner function, \(\tan(x^3)\). This indicates we will need to apply both the chain rule and the power rule.

Step 2: Apply the Chain Rule

To differentiate \(f(x) = \tan^4(x^3)\), we use the chain rule. Let \(u = \tan(x^3)\), then \(f(x) = u^4\). The derivative of \(f(x)\) with respect to \(u\) is \(4u^3\).

Step 3: Differentiate the Inner Function

Continuing with \(u = \tan(x^3)\), find the derivative \(\frac{du}{dx}\). The derivative of \(\tan(x^3)\) is \(\sec^2(x^3)\) multiplied by the derivative of the inner function \(x^3\), which is \(3x^2\). Thus, \(\frac{du}{dx} = 3x^2 \cdot \sec^2(x^3)\).

Step 4: Combine Using Chain Rule

Combine the results from Steps 2 and 3: \(f'(x) = 4u^3 \cdot \frac{du}{dx}\). Substitute back \(u = \tan(x^3)\) to get \(4(\tan^3(x^3)) \cdot 3x^2 \cdot \sec^2(x^3)\).

Step 5: Simplify the Expression

Simplify the expression to find \(f'(x)\). Multiply the constants and combine the terms: \(f'(x) = 12x^2 \tan^3(x^3) \sec^2(x^3)\).

Key Concepts

Chain rule in calculus

The chain rule is an essential technique in calculus that allows us to differentiate composite functions.
When you have a function composed of two functions, say an outer function and an inner function, the chain rule helps you find the derivative efficiently.
In simpler terms, it helps you "chain" the derivatives of these functions together. Here's how it works:

• Identify the outer and inner functions. For our exercise, the outer function is raising something to the fourth power, and the inner function is the tangent of something.
• Differentiate the outer function while keeping the inner function unchanged. Then multiply this by the derivative of the inner function.

Let's take our function: \(f(x) = \tan^4(x^3)\).
To apply the chain rule, set \(u = \tan(x^3)\), making the function \(f(x) = u^4\).
The derivative of \({u^4}\) with respect to \(u\) is \(4u^3\).
Then, find the derivative of \(u\) with respect to \(x\), which involves differentiating \({\tan(x^3)}\).
Finally, multiply these derivatives to find the complete derivative of the original function.
This is how the chain rule helps simplify the differentiation process for complex functions.

Power rule differentiation

The power rule is a fundamental differentiation rule used when differentiating functions of the form \(x^n\).
This rule states that if you have a power of an expression, you can bring the power down as a coefficient and reduce the power by one.
This makes finding derivatives quick and straightforward when dealing with polynomial-like expressions.

• For a function \({u^n}\), the derivative with respect to \(u\) is \(nu^{n-1}\).
• It applies directly but also helps in combination with other rules, like the chain rule.

In the exercise, after identifying \(u = \tan(x^3)\), the function became \(u^4\).
We used the power rule to differentiate it as \(4u^3\).
This determined the derivative of the outer function part, setting the stage for the chain rule's completion.
Combining the power rule with the chain rule, we tackled the original exercise's composite structure to find \(f'(x) = 12x^2 \tan^3(x^3) \sec^2(x^3)\).
The power rule was instrumental in determining the derivative of the outer function before applying the chain rule to the inner function.

Trigonometric differentiation

Trigonometric differentiation involves finding derivatives of trigonometric functions such as sine, cosine, and tangent.
These functions have specific differentiation rules which are crucial for solving calculus problems involving trigonometric expressions.
Let's look at the trig function in this exercise:

• The derivative of \(\tan(x)\) is \(\sec^2(x)\).
• When differentiating \(\tan(x^3)\), it's essential to account for the \(x^3\) inside the tangent, requiring the chain rule.

In our example, differentiating \(\tan(x^3)\) begins with knowing that the basic derivative of \(\tan(x)\) is \(\sec^2(x)\), but here we also need to multiply by the derivative of \(x^3\), which is \(3x^2\).
This leads to the combined derivative, \(3x^2 \sec^2(x^3)\).
Understanding trigonometric differentiation and how to apply chain rule enhances the ability to differentiate more complex functions correctly.
Trigonometric derivatives are foundational in calculus, and knowing these rules helps solve many real-world problems involving oscillations, waves, and rotations.