Question

Differentiate the following functions: \(u=\sqrt[3]{1-x}\)

Short answer

Answer: The derivative of the given function is \(u'=-\frac{1}{3}(1-x)^{-\frac{2}{3}}\).

Step-by-step solution

Identify the outer and inner functions

In this case, the outer function is \(f(u)=u^{\frac{1}{3}}\) and the inner function is \(g(x)=1-x\). The chain rule states that the derivative of a composite function \(f(g(x))\) is: \((f\circ g)'(x) = f'(g(x)) \cdot g'(x)\). In our case, we will find the derivative of the outer function with respect to u and the derivative of the inner function with respect to x.

Find the derivative of the outer function with respect to u

Using the power rule, which states that the derivative of \(f(u)=u^n\) is \(f'(u)=n\cdot u^{n-1}\), we find the derivative of the outer function with respect to u: \(f'(u) = \frac{1}{3}u^{-\frac{2}{3}}\)

Find the derivative of the inner function with respect to x

The inner function is \(g(x)=1-x\). Its derivative with respect to x is: \(g'(x) = -1\)

Apply the chain rule to find the derivative of the composite function

According to the chain rule, the derivative of the composite function is \((f\circ g)'(x) = f'(g(x)) \cdot g'(x)\). Substitute the expressions we found in steps 2 and 3 into this formula: \(f'(g(x)) = \frac{1}{3}(1-x)^{-\frac{2}{3}}\) and \(g'(x) = -1\). Therefore, \((f\circ g)'(x) = \frac{1}{3}(1-x)^{-\frac{2}{3}}\cdot(-1)\)

Simplify the derivative

Now, we will simplify the expression for the derivative: \((f\circ g)'(x) = -\frac{1}{3}(1-x)^{-\frac{2}{3}}\) So, the derivative of the given function is: \(u'=-\frac{1}{3}(1-x)^{-\frac{2}{3}}\)

Key concepts

Chain Rule

When dealing with complex functions, especially those involving multiple nested functions, the Chain Rule becomes a crucial mathematical tool. The Chain Rule allows us to differentiate composite functions effectively. To better understand this, imagine you're peeling layers of an onion, where each layer represents a different function.

• The Chain Rule formula states that if we have a composite function like \(f(g(x))\), the derivative is given by: \((f\circ g)'(x) = f'(g(x)) \cdot g'(x)\).
• This implies that you first differentiate the outer function \(f\) with respect to \(g\), considering \(g\) as your variable and multiply it by the derivative of the inner function \(g\).
• It's essential to identify these inner and outer functions correctly to apply the rule effectively.

Learning to identify \(f\) and \(g\) correctly is key to mastering the Chain Rule. This technique is invaluable as functions get more complicated and layers increase.

Power Rule

The Power Rule is one of the fundamental rules of differentiation, which allows us to easily find the derivative of a power function. This rule states that if a function is in the form of \(u^n\), its derivative is expressed as \(f'(u) = n \cdot u^{n-1}\).

• In our given problem, the outer function is \(f(u) = u^{\frac{1}{3}}\).
• By applying the Power Rule, we find the derivative \(f'(u) = \frac{1}{3}u^{-\frac{2}{3}}\).
• The exponent decreases by one, and you multiply by the original power, making this relatively straightforward compared to other rules.

The Power Rule simplifies the differentiation process of polynomial expressions, ensuring we navigate through polynomial and power functions effortlessly.

Composite Function

A composite function is created when one function is applied to the results of another, effectively combining them into one. This is a common occurrence in calculus and crucial to understanding how functions interact.

• For example, consider our function \(u = \sqrt[3]{1-x}\). Here, \(u\) is defined as a function of \(1-x\), which itself is another function.
• The outer function in this case is \(u^{\frac{1}{3}}\), and the inner function is \(1-x\).
• This layered function setup makes the composite functions ripe for applying the Chain Rule for differentiation.

Composite functions help in modeling real-world scenarios where results depend on multiple varying factors. Mastering this concept leads to deeper insights into complex relationships between variables.