Problem 6 - Calculus

Question

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ \left(x^{2}+1\right)^{2} $$

Step-by-Step Solution

Answer

The derivative is \( 4x^3 + 4x \).

Step 1: Recognize the Function Form

The function given is \( (x^2 + 1)^2 \). It's in the form of a power of a function, which suggests using the chain rule for differentiation.

Step 2: Apply the Chain Rule

The chain rule states that if you have a function of the form \( [u(x)]^n \), the derivative is \( n imes [u(x)]^{n-1} imes u'(x) \). Here, \( u(x) = x^2 + 1 \) and \( n = 2 \).

Step 3: Differentiate the Outer Function

Differentiate \( (x^2 + 1)^2 \) using the chain rule. The derivative is: \[ 2 imes (x^2 + 1)^{2-1} imes \text{Derivative of } (x^2 + 1) \] which simplifies to: \[ 2 imes (x^2 + 1) imes u'(x) \]

Step 4: Differentiate the Inner Function

Find the derivative of \( u(x) = x^2 + 1 \). The derivative of \( x^2 \) is \( 2x \), and the derivative of a constant is 0. Therefore, \( u'(x) = 2x \).

Step 5: Combine the Results

Substitute \( u'(x) = 2x \) back into the chain rule expression: \[ 2 imes (x^2 + 1) imes 2x \] which simplifies to: \[ 4x imes (x^2 + 1) \]

Step 6: Final Simplification

Expand the expression to obtain the final derivative: \[ 4x(x^2 + 1) = 4x^3 + 4x \]

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used when differentiating composite functions. A composite function is one that is created by nesting one function inside another, often expressed as \( f(g(x)) \). The purpose of the chain rule is to help us deal with these layered functions efficiently.

• The basic formula for the chain rule is: If \( y = [u(x)]^n \), then the derivative \( y' = n \times [u(x)]^{n-1} \times u'(x) \).
• This formula shows that we differentiate the outer function and multiply it by the derivative of the inner function.

By understanding and applying the chain rule, solving differentiation problems involving composite functions becomes much more manageable.

Power Rule

The power rule is one of the simplest yet most powerful rules for finding derivatives. It simplifies the process of differentiating functions of the form \( x^n \), where \( n \) could be any real number, including integers and fractions.

• The formula for the power rule is: \( \frac{d}{dx}x^n = nx^{n-1} \).
• This means that the new exponent of \( x \) is one less than the original, and we multiply by the original exponent.

The power rule is particularly useful in our example exercise because it is applied to both the main function and when differentiating \( u(x) = x^2 + 1 \).

Differentiation

Differentiation is the core concept in calculus that involves finding the rate at which a function is changing at any given point. In simple terms, it's the process of finding the derivative of a function.

• A derivative represents the slope of the tangent line to the function at a particular point.
• The notation for a derivative is \( f'(x) \) or \( \frac{dy}{dx} \).

In the exercise, differentiation helps find the rate of change of the function \( (x^2 + 1)^2 \), resulting in the function's slope at any point \( x \). This is a vital tool not only in mathematics but also in physics, engineering, and other disciplines needing to measure change.

Inner Function

The inner function is the function within the composite function that is differentiated using the chain rule. In our exercise, the inner function is \( u(x) = x^2 + 1 \).

• Identifying the inner function is crucial as it is needed to apply the chain rule correctly.
• Once identified, we find its derivative separately.

In this case, after identifying \( u(x) = x^2 + 1 \), we find its derivative, \( u'(x) = 2x \), which is instrumental in computing the final derivative of the outer function. Recognizing the inner function is a key step in the chain rule process.

Outer Function

The outer function in differentiation using the chain rule is the function that "wraps around" the inner function. It's the function you'll differentiate first in the chain rule process.

• In our example, the outer function is \( (x^2 + 1)^2 \).
• When using the chain rule, you differentiate the outer function while keeping the inner function intact, resulting in \( 2(x^2+1)^{1} \).

Understanding the role of the outer function is vital, as it sets the structure for how the inner function's derivative gets incorporated into the total derivative. This overlay helps in systematically breaking down the function for easy differentiation.