Question

Differentiate the following funetions: \(u=x^{3}(1-x)^{4}\)

Short answer

Question: Find the derivative of the function \(u = x^3(1-x)^4\). Answer: \(u' = 3x^2(1-x)^4 - 4x^3(1-x)^3\).

Step-by-step solution

Identify the two functions

In this problem, we have \(f(x) = x^3\) and \(g(x) = (1-x)^4\).

Differentiate f(x)

We apply the power rule which states that the derivative of \(x^n\) is \(nx^{n-1}\). So the derivative of \(f(x) = x^3\) is \(f'(x) = 3x^2\).

Differentiate g(x)

We use the chain rule on \(g(x) = (1-x)^4\). The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is \(u^4\) and the inner function is \(1-x\). The derivative of the outer function is \(du/dt = 4u^3\) and the derivative of the inner function is \(d(1-x)/dx = -1\). So, applying the chain rule, we have \(g'(x) = 4(1-x)^3(-1) = -4(1-x)^3\).

Apply the product rule

Now we apply the product rule to find the derivative of the function u. That is, we compute \(u' = f'(x)g(x) + f(x)g'(x)\). So, we have: \(u' = 3x^2(1-x)^4 + x^3(-4(1-x)^3)\).

Simplify the result

Let's simplify the expression for \(u'\): \(u' = 3x^2(1-x)^4 - 4x^3(1-x)^3\). This is the final expression for the derivative of the function u.

Key concepts

Power Rule

The power rule is a fundamental tool in calculus used to differentiate functions of the form \( f(x) = x^n \), where \( n \) is a constant. This simple yet powerful rule states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \).

• For example, if you have \( f(x) = x^3 \), applying the power rule gives \( f'(x) = 3x^2 \).
• This rule is applicable in situations when a term is just a variable raised to a constant power.

Remember, the power rule only applies directly to terms like \( x^n \). Special care is required when dealing with more complex expressions where additional rules like chain or product rule might be necessary.

Chain Rule

The chain rule is essential for finding the derivative of composite functions. A composite function is one where you have an outer function and an inner function. The chain rule states:

• To differentiate \( g(x) = (1-x)^4 \), think of it as \( (u(x))^4 \) with \( u(x) = 1 - x \).
• First, differentiate the outer function: \( 4(u(x))^3 \).
• Then, differentiate the inner function: the derivative of \( (1-x) \) is \( -1 \).

When combining these, you multiply the derivative of the outer function by the derivative of the inner function: \( g'(x) = 4(1-x)^3(-1) = -4(1-x)^3 \). This approach simplifies the process of dealing with nested functions by breaking them down step by step.

Product Rule

The product rule comes into play when differentiating functions that are multiplied together. For instance, in the expression \( u(x) = x^3(1-x)^4 \), we need to apply the product rule.The rule states that if you have a product of two functions \( f(x) \) and \( g(x) \), the derivative \( u'(x) \) is given by:\[ u'(x) = f'(x)g(x) + f(x)g'(x) \] To apply this to our example:

• First, differentiate \( f(x) = x^3 \) to get \( f'(x) = 3x^2 \).
• Next, use the previously found \( g'(x) = -4(1-x)^3 \).
• Plug these into the formula, resulting in \( u' = 3x^2(1-x)^4 + x^3(-4(1-x)^3) \).

This rule is crucial for handling the differentiation of products of functions and ensures that each part is accounted for in the process.