Problem 63 - Calculus and its applications

Question

Let \(f(u)=u^{3}\) and \(g(x)=u=2 x^{4}+1\) Find \((f \circ g)^{\prime}(-1)\)

Step-by-Step Solution

Answer

\((f \circ g)^{\prime}(-1) = -216\).

Step 1: Express Composite Function

The composite function \((f \circ g)(x)\) is defined as \(f(g(x))\). Substituting \(g(x)\) into \(f(u)\), we get \(f(g(x)) = (2x^4 + 1)^3\).

Step 2: Differentiate the Composite Function

To find the derivative of the composite function, \((f \circ g)^{\prime}(x)\), use the chain rule. The outer function \(f(u) = u^3\) has a derivative of \(3u^2\). The derivative of the inner function \(g(x)=2x^4 + 1\) is \(8x^3\). Thus, the derivative is \((f \circ g)^{\prime}(x) = 3(2x^4 + 1)^2 \cdot 8x^3\).

Step 3: Simplify the Derivative Expression

Simplify the expression obtained from the chain rule:\[(f \circ g)^{\prime}(x) = 24x^3(2x^4 + 1)^2.\]

Step 4: Evaluate the Derivative at x = -1

Substitute \(x = -1\) into the derivative expression:\[(f \circ g)^{\prime}(-1) = 24(-1)^3(2(-1)^4 + 1)^2.\]First calculate \((-1)^3 = -1\) and \(2(-1)^4 + 1 = 3\).Then evaluate \((3)^2 = 9\), so:\[(f \circ g)^{\prime}(-1) = 24(-1)(9) = -216.\]

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus, especially when dealing with composite functions. It allows us to differentiate a function composed of two or more functions. Imagine you have a chain of functions: an outer function and one or more inner functions. In our exercise, we have two functions: the outer function, \(f(u) = u^3\), and the inner function, \(g(x) = 2x^4 + 1\).
The chain rule helps us find the derivative of the composite function \((f \circ g)(x)\) by taking the derivative of the outer function with respect to the inner function, then multiplying it by the derivative of the inner function with respect to \(x\). In formula form, it's expressed as:

• \((f \circ g)^{7}(x) = f'(g(x)) \cdot g'(x)\).

This systematic approach offers a way to navigate through layers of functions, making it easier to crack even the most intricate compositions.

Derivative Evaluation

Once the derivative of the composite function is obtained, evaluating it at a specific point provides insights into the behavior of the function at that location. In our step-by-step solution, we've derived that the derivative \((f \circ g)^{7}(x) = 24x^3(2x^4 + 1)^2\).
To evaluate it at \(x = -1\), we replace \(x\) with \(-1\) in the expression and carefully calculate:

• \((-1)^3 = -1\),
• \(2(-1)^4 + 1 = 3\).
• Consequently, \((3)^2 = 9\).
• Thus, \(24(-1)(9) = -216\).

This evaluation process, step by step, reveals the derivative's value at the specified point, offering insight into the function's slope and direction at \(x = -1\).

Function Differentiation

Function differentiation involves finding the derivative of a function, reflecting how the function changes as its inputs change. In our exercise, the focus is on differentiating a composite function. Each component of the composite must be differentiated:
1. **Outer function differentiation**: The outer function is \(f(u) = u^3\), with a derivative of \(3u^2\). This represents the change of the outer function based on changes in \(u\).
2. **Inner function differentiation**: The inner function is \(g(x) = 2x^4 + 1\), with a derivative of \(8x^3\). This derivative indicates how \(g(x)\) varies as \(x\) changes, serving as a bridge component in our differentiated composite function.
By applying chain rule and finding these derivatives, we achieve a complete understanding of how \(f(g(x))\) evolves. Mastering function differentiation is crucial for anyone studying calculus as it forms the backbone of analyzing function behavior and dynamics.