Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Differentiate \(f(x)=5 \tan \left(x^{2}\right)+\arctan 3 x\). (b) Find \(\int 10 x \sec ^{2}\left(x^{2}\right)+\frac{3}{1+9 x^{2}} d x\)

Step-by-Step Solution

Answer

(a) The derivative is \(f'(x) = 10x \cdot \sec^2(x^2) + \frac{3}{1+(3x)^2}\) \n (b) The integral is \( \int 10 x \sec ^{2}(x^{2})+\frac{3}{1+9 x^{2}} dx = 5 \tan(x^2) + \arctan(3x) + C\)

Step 1: Differentiate the function for (a)

The function we would like to differentiate is \(f(x)=5 \tan (x^{2})+\arctan 3x\). We have to use the Chain Rule for finding derivative of composite functions. Differentiate term by term using the Derivative Rule. The derivative of \(\tan u\) is \(\sec^2 u\) multiplied by the derivative of \(u\), and the derivative of \(\arctan v\) is \(1/(1+v^2)\) multiplied by the derivative of \(v\). So, after applying these rules, the derivative is \(f'(x) = 5 \cdot 2x \cdot \sec^2(x^2) + \frac{3}{1+(3x)^2}\).

Step 2: Integral of the derivative function for (b)

We need to integrate the function we got in the previous step, \(10x \sec^2(x^2) + \frac{3}{1+9x^2}\). The first term is the integral of \(10x \sec^2(x^2)\) dx which is simply \( 5 \tan(x^2)\), because the derivative of the tangent function with respect to its inner function is \( \sec^2\) of the inner function multiplied by derivative of inner function. The second term is the integral of \(\frac{3}{1+9x^2} dx\), which is \(\arctan(3x)\), because the derivative of the arctangent function is \(\frac{1}{1+x^2}\). Sum these two terms: then the integral of the whole function would be: \( 5 \tan(x^2) + \arctan(3x) + C\), where \(C\) is the constant of integration.

Step 3: End of Task (b)

The task (b) was to find the indefinite integral of the derivative function, and we did that in the previous step. Don't forget, it's important to always include the constant of integration \(C\) in indefinite integrals.

Key Concepts

Derivative

In calculus, differentiation refers to the process of finding the derivative of a function. A derivative represents the rate of change of a function concerning one of its variables.
For instance, in the given exercise, we are asked to differentiate the function: \( f(x) = 5 \tan(x^2) + \arctan(3x) \). Differentiation helps us understand how a function changes, which is crucial in various applications, such as physics and engineering.To differentiate the given function, we need to apply specific rules:

• The derivative of \( \tan(u) \) is \( \sec^2(u) \), multiplied by the derivative of \( u \) according to the chain rule.
• The derivative of \( \arctan(v) \) is \( \frac{1}{1+v^2} \), again multiplied by the derivative of \( v \).

By employing these rules, we break down the differentiation into manageable parts. For \( 5 \tan(x^2) \), we differentiate the inner \( x^2 \) first to find \( 2x \), leading to \( 5 \cdot 2x \cdot \sec^2(x^2) \). Similarly, for \( \arctan(3x) \), the derivative results in \( \frac{3}{1+(3x)^2} \), after applying the chain rule.Thus, the derivative of the entire function is: \( f'(x) = 5 \cdot 2x \cdot \sec^2(x^2) + \frac{3}{1+(3x)^2} \).

Integral

In mathematics, integration is the reverse process of differentiation. It is used to find the function given its rate of change, known as the anti-derivative. Integration helps in calculating the area under curves, among other applications.
In the provided exercise, the goal is to integrate the function \( 10x \sec^2(x^2) + \frac{3}{1+9x^2} \), which we determined during differentiation.Integrating can be simplified by understanding that it reverses the steps of differentiation:

• For \( 10x \sec^2(x^2) \), recognize that it resembles the derivative of \( \tan(u) \). When integrated, it results in \( 5 \tan(x^2) \).
• For \( \frac{3}{1+9x^2} \), notice that this is similar to the derivative leading to \( \arctan(3x) \). Thus, the integral is \( \arctan(3x) \).

These observations allow us to bring together the integrals of both terms. Therefore, combining the integrals gives us: \( \int \big( 10x \sec^2(x^2) + \frac{3}{1+9x^2} \big)\, dx = 5 \tan(x^2) + \arctan(3x) + C\), where \( C \) is the constant of integration that accounts for any added constant during differentiation.

Chain Rule

Understanding the chain rule is vital in calculus, especially when dealing with composite functions. The chain rule allows us to differentiate compositions by breaking them up into simpler parts.
This rule states that to differentiate a composite function \( f(g(x)) \), you multiply the derivative of the outer function by the derivative of the inner function.In the exercise, consider the term \( \tan(x^2) \). It's a composite of \( \tan(u) \) where \( u = x^2 \). The chain rule tells us to:

• Differentiate \( \tan(u) \) to obtain \( \sec^2(u) \).
• Multiply by the derivative of \( u \), which is \( 2x \).

By applying the chain rule, the differentiation of \( \tan(x^2) \) results in \( 2x \sec^2(x^2) \), illustrating the power of the chain rule in simplifying complex differentiation processes.
Similarly, for \( \arctan(3x) \), the chain rule is employed. The outer function, \( \arctan(v) \), gives \( \frac{1}{1+v^2} \) when differentiated. The inner function, with \( 3x \), differentiates to \( 3 \), linking them through multiplication.In conclusion, the chain rule effectively handles nested functions, making it a fundamental tool in calculus for tackling derivatives.