Problem 8 - Calculus - AP Edition

Question

Find \(f^{\prime}(x)\) $$f(x)=\left(3 x^{2}+2 x-1\right)^{6}$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 6(3x^2 + 2x - 1)^5(6x + 2) \).

Step 1: Identify the Function Form

The function is of the form \( u^n \) where \( u = 3x^2 + 2x - 1 \) and \( n = 6 \). This means we'll use the chain rule to differentiate it.

Step 2: Apply the Chain Rule

The chain rule states that if \( f(x) = (u(x))^n \), then \( f'(x) = n(u(x))^{n-1} \cdot u'(x) \). Here, we differentiate the outer function \( u^6 \) followed by the inner function \( u = 3x^2 + 2x - 1 \).

Step 3: Differentiate the Outer Function

The derivative of \( u^6 \) with respect to \( u \) is \( 6u^5 \). So we have \( 6(3x^2 + 2x - 1)^5 \).

Step 4: Differentiate the Inner Function

Find the derivative of \( u = 3x^2 + 2x - 1 \). The derivative, \( u'(x) \), is \( 6x + 2 \).

Step 5: Multiply the Derivatives

Combine the results from the previous steps using the chain rule: \( f'(x) = 6(3x^2 + 2x - 1)^5 \cdot (6x + 2) \).

Step 6: Final Expression for f'(x)

The derivative of \( f(x) = (3x^2 + 2x - 1)^6 \) is given by \( f'(x) = 6(3x^2 + 2x - 1)^5 \cdot (6x + 2) \).

Key Concepts

Differentiation

Differentiation is one of the core concepts in calculus. It is the mathematical process of finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point. The fundamental idea is to calculate the slope of the tangent line to the function at a particular point. This slope provides us with valuable information about the behavior of the function.To differentiate a function, you essentially find out how much the function value changes for a small change in input value. It allows you to measure:

• Rate of change
• Velocity and acceleration in physics
• Gradient in geometric figures

In our exercise involving \(f(x) = (3x^2 + 2x - 1)^6\), differentiation enables us to determine the slope of the function at any point along its curve. Thus, helping in sketching the graph and understanding its growth or decay over intervals.

Power Rule

The power rule is a basic rule of differentiation used to find the derivative of a function of the form \(x^n\), where \(n\) is any real number. This rule states that the derivative of \(x^n\) is \((nx^{n-1})\).The power rule simplifies the process of differentiation. Here's how it works:

• Multiply the original exponent by the coefficient in front of the \(x\).
• Subtract one from the original exponent to form the new exponent.

In the context of our exercise, the expression \(u^6\) in \(f(x) = (3x^2 + 2x - 1)^6\) uses the power rule when differentiating the outer function, resulting in \(6u^5\). This step is crucial when applying further differentiation rules, like the chain rule, to more complex functions.

Derivative Computation

Derivative computation involves more than just applying simple rules. It often requires a combination of techniques, especially when dealing with complex composite functions like in this exercise.The chain rule is essential in this context. When you have a function composed of other functions, such as \(f(x) = (g(x))^n\), the chain rule helps differentiate it. The formula for the chain rule is \( f'(x) = n(g(x))^{n-1} \cdot g'(x) \). This rule enables us to calculate the derivative of the composite expression \( (3x^2 + 2x - 1)^6 \).Here's a breakdown of how derivative computation was applied in the solution:

• Differentiating the outer function \(u^6\) using the power rule gave \(6(3x^2 + 2x - 1)^5\).
• Finding the derivative of the inner function \(u = 3x^2 + 2x - 1\) resulted in \(6x + 2\).
• Multiplying these results provided the final derivative \(f'(x) = 6(3x^2 + 2x - 1)^5 \times (6x + 2)\).

Understanding derivative computation not only helps in solving such exercises but also gives insight into the mathematical relationships within complex expressions.