Question

a. Calculate \(\frac{d}{d x}\left(x^{2}+x\right)^{2}\) using the Chain Rule. Simplify your answer. b. Expand \(\left(x^{2}+x\right)^{2}\) first and then calculate the derivative. Verify that your answer agrees with part (a).

Short answer

Question: Find the derivative of the composite function \({(x^2 + x)}^{2}\) using the Chain Rule and by expanding the function, then compare the results to verify they are equal. Answer: The derivative of the composite function using the Chain Rule is \(2(x^2+x)(2x+1)\). After expanding the function and finding its derivative, we get \(4x^3+6x^2+2x\), which can be factored to \(2x(x^2+3x+1)\) and further to \(2x(x+1)(2x+1)\). Comparing the two results, we can verify that they are equal.

Step-by-step solution

Outer Function

The outer function is \(f(u)=u^2\) , where \(u=x^2+x\).

Inner Function

The inner function is \(g(x)=x^2+x\). Step 2:Apply the chain rule

Find the Derivative of the Outer Function

The derivative of outer function, \(f'(u)=2u\).

Find the Derivative of the Inner Function

The derivative of inner function, \(g'(x)=2x+1\). Step 3: Apply Chain Rule

Derivative of Composite function using Chain Rule

Derivative of the composite function, \(\frac{d}{dx}[(x^2+x)^2]=f'(g(x))\cdot g'(x) = 2(x^2+x)(2x+1)\). The derivative simplifies to \(2(x^2+x)(2x+1)\). #Phase 2#: Expanding and finding the derivative Step 4:Expand the function

Expansion

Expanding \({(x^2+x)}^2 = (x^2+x)(x^2+x) = x^4+2x^3+x^2\). Step 5: Calculate the derivative

Derivative of the Expanded Function

The derivative of the expanded function, \(\frac{d}{dx}(x^4+2x^3+x^2)=4x^3+6x^2+2x\). Step 6: Compare the results

Verification

Rewrite the expanded derivative in factored form: \(2x(x^2+3x+1)\). Factor the result found in Phase 1: \(2(x^2+x)(2x+1)=2x(x+1)(2x+1)\), so we can see that the derivatives are equal after expanding and factoring them out. So, the derivatives from part (a) and part (b) agree.

Key concepts

Derivative

Derivatives are a fundamental concept in calculus, representing the rate at which a function changes. When we take the derivative of a function, we are essentially finding the slope of the tangent line to the graph of the function at any given point.

In the given exercise, we are tasked with finding the derivative of the composite function \((x^2 + x)^2\).

Using the Chain Rule, this involves identifying both the inner and outer functions. The outer function is \(f(u) = u^2\) and the inner function is \(g(x) = x^2 + x\).

We then differentiate each function separately:

• The derivative of the outer function is \(2u\).
• The derivative of the inner function is \(2x + 1\).

Applying the Chain Rule, the derivative of the composite function becomes the product of these derivatives:
\[(f'(u) \cdot g'(x)) = 2(x^2+x)(2x+1)\]

This process highlights the beauty of the Chain Rule in simplifying the differentiation of composite functions.

Composite Function

A composite function is a function that is made by combining two or more functions. This is akin to layering one function over another, leading to a new function that often has more complex behavior.

In calculus, understanding composite functions allows us to handle more advanced differentiation using rules like the Chain Rule.

In our exercise, we deal with the composite function \((x^2 + x)^2\).
Here:

• The inner function is \(g(x) = x^2 + x\).
• The outer function is \(f(u) = u^2\), where \(u = x^2 + x\).

The significance of composite functions is that they let us apply the Chain Rule, which 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.

This highlights how we can effectively tackle complicated functions by breaking them down into simpler parts, differentiating each, and then combining the results appropriately.

Polynomial Expansion

Polynomial expansion involves expressing a power or multiplication of polynomials as a sum of terms. This entails multiplying each part of the polynomial to simplify or analyse the expression more efficiently.

In our problem, we expanded the function \( (x^2+x)^2 \) by multiplying \( (x^2+x) \) by itself. This results in \( x^4 + 2x^3 + x^2 \).

Expanding a polynomial can make differentiation easier, especially if dealing with functions composed of multiple polynomial terms. Once expanded, we can take the derivative term by term:

• The derivative of \( x^4 \) is \( 4x^3 \).
• The derivative of \( 2x^3 \) is \( 6x^2 \).
• The derivative of \( x^2 \) is \( 2x \).

The complete derivative is now \( 4x^3 + 6x^2 + 2x \), simplifying verification of the solution obtained via other methods, like the Chain Rule.

This expansion allows us to verify the agreement between different approaches to finding derivatives, ensuring accuracy and understanding of various calculus techniques.