2.3 Q-1

Question Statement

Differentiate the following expression with respect to $x$:
$Y = x^4 + 2x^3 + x^2$

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Background and Explanation

In this problem, we will use the power rule of differentiation, which states:
$\frac{d}{dx} \left[ x^n \right] = n \cdot x^{n-1}$
This rule allows us to quickly find the derivative of polynomial terms by multiplying the power of $x$ with its coefficient and reducing the power by 1.

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Solution

Let the given function be: $Y = x^4 + 2x^3 + x^2$

Step 1: Differentiate each term

We will apply the power rule to each term of the polynomial:

1. For $x^4$:
   $\frac{d}{dx} \left[ x^4 \right] = 4x^3$

2. For $2x^3$:
   $\frac{d}{dx} \left[ 2x^3 \right] = 6x^2$

3. For $x^2$:
   $\frac{d}{dx} \left[ x^2 \right] = 2x$

Step 2: Combine the derivatives

Adding all the individual derivatives together: $\frac{dy}{dx} = 4x^3 + 6x^2 + 2x$

Final Answer:
$\frac{dy}{dx} = 4x^3 + 6x^2 + 2x$

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Key Formulas or Methods Used

• Power Rule:
  $\frac{d}{dx} \left[ x^n \right] = n \cdot x^{n-1}$

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Summary of Steps

1. Identify the given function:
   $Y = x^4 + 2x^3 + x^2$

2. Apply the power rule to differentiate each term:

   • $\frac{d}{dx} \left[ x^4 \right] = 4x^3$
   • $\frac{d}{dx} \left[ 2x^3 \right] = 6x^2$
   • $\frac{d}{dx} \left[ x^2 \right] = 2x$

3. Combine the results to find the derivative:
   $\frac{dy}{dx} = 4x^3 + 6x^2 + 2x$

This gives the rate of change of the function $Y$ with respect to $x$.