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Chapter 2: Problem 6

Find the derivative of each function. $$f(x)=\left(x^{3}+x-1\right)^{3}$$

Short Answer

Expert verified
Therefore, the derivative of the function \(f(x)= (x^{3}+x-1)^{3} \) is \( f'(x)=3(x^{3}+x-1)^{2} \cdot (3x^{2}+1) \).

Step by step solution

01

Identify the outer and inner functions

Recognize the given function as a composition of two functions. The inner function, \(g(x)\), is \(x^{3}+x-1\) and the outer function, \(f(u)\), is \(u^{3}\). We write our function \(f(x)\) as \(f(g(x))\) where \(f(u)= u^{3}\) and \(g(x)=x^{3}+x-1 \).
02

Compute the derivatives of the outer and inner functions

We find the derivative of the inner function, \(g'(x)=3x^{2}+1\) and the derivative of the outer function, \(f'(u)= 3u^{2}\).
03

Apply the chain rule

The chain rule, which is \(f'(g(x)) \cdot g'(x)\), gives us the derivative of the given function. Substitute \(g(x)\) as \(u\) in \(f'(u)\) to get \(f'(g(x))= 3g(x)^{2}\). The derivative of the given function is hence \(f'(x)= f'(g(x)) \cdot g'(x)= 3g(x)^{2} \cdot g'(x) \) \( = 3(x^{3}+x-1)^{2} \cdot (3x^{2}+1) \). observe that \(g(x)= x^{3}+x-1\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differentiation
Differentiation is a cornerstone concept in calculus, which involves computing the derivative of a function. The derivative represents the rate at which a function's value changes with respect to changes in its input value. In essence, it tells us how a function
Composite Functions
Understanding composite functions is crucial when dealing with more complex scenarios in calculus. A composite function is created when one function is applied to the result of another function. Mathematically, if you have two functions, say f and g, the composite function f(g(x)) is formed by applying g to x, and then applying f to the result of g(x). This can be visually imagined as a process where the output of one function becomes the input for the next function.
Chain Rule
The chain rule is a powerful tool in differentiation when dealing with composite functions. It enables us to differentiate a function of a function, so to speak. If we have a function h(x) = f(g(x)), then the derivative h'(x) is found by multiplying the derivative of the outer function f at g(x), denoted as f'(g(x)), with the derivative of the inner function g, denoted g'(x). The beauty of the chain rule is that it breaks down seemingly intimidating tasks into manageable bites by allowing us to take derivatives step by step.
Calculus
Calculus, the mathematical study of continuous change, is a branch of mathematics that deals with concepts such as limits, functions, derivatives, integrals, and infinite series. It has two main branches: Differential Calculus, concerning rates of change and slopes of curves; and Integral Calculus, concerning accumulation of quantities and the areas under curves. Together, these disciplines are employed to solve problems involving dynamic systems, geometry, engineering, and more. Knowing how to work through calculus operations, including using the chain rule for differentiation, empowers students to tackle a wide range of mathematical challenges.

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