Question

Differentiate the following funetions: \(u=\frac{x}{(1-x)^{3}}\)

Short answer

Answer: The derivative of the given function is \(u'=\frac{(1-x)^{2}(1+2x)}{(1-x)^{6}}\).

Step-by-step solution

Identify the numerator and denominator

In this function, we have the numerator \(v=x\) and the denominator \(w=(1-x)^{3}\).

Find the derivatives of the numerator and denominator

To find the derivatives of the numerator and denominator, we will use the chain rule and the power rule. For the numerator: The derivative of \(v=x\) with respect to \(x\) is just \(1\). So \(v'=1\). For the denominator: The derivative of \(w=(1-x)^{3}\) is given by the chain rule: \(w' = 3(1-x)^{2}\cdot(-1) = -3(1-x)^{2}\)

Apply the quotient rule

Now we will apply the quotient rule to find the derivative of the entire function: \(u'=\frac{v'w-vw'}{w^2}\) Plug in the values from Steps 1 and 2: \(u'=\frac{1\cdot(1-x)^{3}-x\cdot(-3(1-x)^{2})}{(1-x)^{6}}\)

Simplify the expression

Simplify the expression in the numerator: \(\frac{(1-x)^{3}+3x(1-x)^{2}}{(1-x)^{6}}\) Factor out \((1-x)^{2}\) from the numerator: \(\frac{(1-x)^{2}[(1-x) + 3x]}{(1-x)^{6}}\) Combine terms in the parentheses: \(\frac{(1-x)^{2}(1+2x)}{(1-x)^{6}}\)

Write the final answer

The derivative of the given function is: \(u'=\frac{(1-x)^{2}(1+2x)}{(1-x)^{6}}\)

Key concepts

Quotient Rule

The quotient rule is a technique in calculus for finding the derivative of a quotient of two functions. It states that if you have a function \(u\) which is the division of one function \(v\) by another \(w\), the derivative of \(u\) is given by:
\[ u' = \frac{v'w - vw'}{w^2} \]
Where \(v'\) and \(w'\) are the derivatives of \(v\) and \(w\) respectively. Apply this to the given function \(u=\frac{x}{(1-x)^{3}}\), we identify the numerator as \(x\) and the denominator as \( (1-x)^{3}\), then use the quotient rule formula to calculate its derivative. It's vital to first find the derivatives of the numerator and denominator separately, which we will next see involving the chain rule and power rule.

Chain Rule

The chain rule is a fundamental method in calculus for finding the derivative of composite functions. When a function \(w\) is nested within another function, say \(w(\text{g}(x))\), the derivative of the composition with respect to \(x\) is the derivative of the outer function \(w\) evaluated at the inner function \(\text{g}(x)\), multiplied by the derivative of the inner function itself. This is written as:
\[ w'(\text{g}(x)) = w'(\text{g}) \cdot \text{g}'(x) \]
Our example used the chain rule for finding the derivative of the denominator \(w=(1-x)^{3}\), recognizing \(\text{g}(x) = 1-x\) as the inner function and applying the power rule for the outer function which resulted in \(w' = -3(1-x)^{2}\). The chain rule allows us to differentiate complex expressions by breaking them down into their constituent parts.

Power Rule

The power rule is one of the most versatile and basic rules for differentiating functions in calculus, especially when dealing with polynomial expressions. It states that if we have a function \(v = x^n\), where \(n\) is a constant real number, then the derivative of \(v\) with respect to \(x\) is:\
\[ v' = n \cdot x^{n-1} \]
In the exercise provided, we applied the power rule to both the numerator \(x\), which has an implicit power of \(1\), giving us the derivative of \(1' = 1\), and to the denominator \(w = (1-x)^3\). For the denominator, after employing the chain rule, the power rule helped us to ascertain that each term is to be decreased by one in its power and multiplied by its original exponent. This process allows us to differentiate terms with ease and ensures accuracy within polynomial differentiations.