Problem 12 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=\left(x^{3}+1\right)^{4} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 12x^2(x^3 + 1)^3 \).

Step 1: Understand the Generalized Power Rule

The Generalized Power Rule states that if you have a function of the form \( g(x) = (h(x))^n \), where \( h(x) \) is a differentiable function and \( n \) is a constant, then the derivative \( g'(x) \) is \( n(h(x))^{n-1} \cdot h'(x) \). In this exercise, \( f(x) = (x^3+1)^4 \), so \( h(x) = x^3 + 1 \) and \( n = 4 \).

Step 2: Differentiate the Outer Function

Apply the Generalized Power Rule to the outer function \( (x^3 + 1)^4 \). The derivative of \( (x^3 + 1)^4 \) with respect to \( x \) is \( 4(x^3 + 1)^3 \). This is because we bring down the power 4, subtract 1 from the power, resulting in a new exponent of 3.

Step 3: Differentiate the Inner Function

Find the derivative of the inner function, which is \( x^3 + 1 \). The derivative \( h'(x) \) is \( 3x^2 \), since the derivative of \( x^3 \) is \( 3x^2 \) and the derivative of the constant 1 is 0.

Step 4: Apply the Chain Rule

Multiply the derivative of the outer function by the derivative of the inner function. Thus, the derivative of \( f(x) = (x^3 + 1)^4 \) is \( 4(x^3 + 1)^3 \cdot 3x^2 \).

Step 5: Simplify the Expression

Combine and simplify the expression to find the final derivative. \( f'(x) = 4 \cdot 3x^2 \cdot (x^3 + 1)^3 = 12x^2(x^3 + 1)^3 \).

Key Concepts

Derivative

A **derivative** tells us how a function changes as its input changes. It's like having a speedometer for how fast the function's value is moving when you tweak a bit of the input. In more technical terms, the derivative measures the rate of change of a function with respect to a variable. For a function \( f(x) \), its derivative is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).

Calculating a derivative involves understanding various rules and concepts such as the power rule and chain rule. These are ways to simplify complex functions and find their derivatives without going back to the basic definition every time. For instance, if you have a function like \( (x^3+1)^4 \), its derivative will combine multiple rules, making the computation smoother and more intuitive.

Derivatives are fundamental in calculus and appear in everything from physics, where they help model motion, to economics, where they can show cost changes in response to various factors.

Chain Rule

The **chain rule** is a powerful technique in calculus used to find the derivative of a composite function. A composite function is simply a function built from two or more other functions, such as \( f(x) = (x^3 + 1)^4 \). Here, we have an outer function \((u)^4\) and an inner function \(u = x^3 + 1\).

Using the chain rule allows us to differentiate these layered functions systematically. The rule states: if \( y = g(h(x)) \), then the derivative \( \frac{dy}{dx} = g'(h(x)) \cdot h'(x) \). Essentially, you differentiate the outer function first while keeping the inner function unchanged, then multiply by the derivative of the inner function.

Applying this to our example, we first focus on the outer layer \((x^3 + 1)^4\). Derivative of the outer function becomes \(4(x^3 + 1)^3\). Next, the inner function's derivative \(h'(x) = 3x^2\) is computed. Finally, multiplying these results gives us the complete derivative.

Power Rule

The **power rule** is one of the simpler rules for finding derivatives, especially handy when dealing with polynomials. It states that if \( h(x) = x^n \), then the derivative \( h'(x) = nx^{n-1} \). It's like saying: bring down the power as a multiplier and reduce the power by one.

Consider applying the power rule to our exercise with a function \((x^3 + 1)^4\). Although the primary power here involves a combination of functions, the power rule kick-starts our differentiation process. It helps simplify the outer function \((x^3 + 1)^4\) to \(4(x^3 + 1)^3\). This action is straightforward because you're only dealing with powers of \(x\).

The power rule shines its brightest when combined with other calculus tools, making it vital for handling more complex expressions effortlessly.