Question Statement

Find the derivative of the function:

$y=\frac{x^4+2x^3-1}{x^2}$

with respect to $x$.

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Background and Explanation

This problem involves differentiating a rational function where the numerator and denominator are both polynomials. While the expression can be simplified first by dividing each term by $x^2$, we will demonstrate the quotient rule method, which is essential for differentiating ratios of functions that cannot be easily simplified.

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Solution

Given the function:

$y=\frac{x^4+2x^3-1}{x^2}$

We identify the numerator as $u=x^4+2x^3-1$ and the denominator as $v=x^2$. To find $\frac{dy}{dx}$, we apply the quotient rule:

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

First, compute the derivatives of $u$ and $v$ using the power rule:

• $\frac{d}{dx}(x^4+2x^3-1)=4x^3+6x^2+0=4x^3+6x^2$
• $\frac{d}{dx}(x^2)=2x$

Substituting these into the quotient rule formula:

$\frac{dy}{dx}=\frac{x^2\cdot \frac{d}{dx}\left(x^4+2x^3-1\right)-\left(x^4+2x^3-1\right)\frac{d}{dx}\left(x^2\right)}{{\left(x^2\right)}^2}$

$=\frac{x^2\left(4x^3+6x^2-0\right)-\left(x^4+2x^3-1\right)(2x)}{x^4}$

Now expand the numerator by distributing $x^2$ in the first term and $2x$ in the second term:

$=\frac{4x^5+6x^4-2x^5-4x^4+2x}{x^4}$

Combine like terms in the numerator ($4x^5-2x^5=2x^5$ and $6x^4-4x^4=2x^4$):

$=\frac{2x^5+2x^4+2x}{x^4}$

Factor out $2x$ from the numerator:

$\frac{dy}{dx}=\frac{2x\left(x^4+x^3+1\right)}{x^4}$

Cancel the common factor of $x$ from the numerator and denominator:

$\frac{dy}{dx}=\frac{2\left(x^4+x^3+1\right)}{x^3}$

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Key Formulas or Methods Used

• Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$
• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Derivative of a Constant: $\frac{d}{dx}(c)=0$

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Summary of Steps

1. Identify the numerator $u$ and denominator $v$ of the rational function
2. Differentiate $u$ and $v$ separately using the power rule (remembering that the derivative of $-1$ is $0$)
3. Apply the quotient rule formula: $\frac{v\cdot u^{\prime }-u\cdot v^{\prime }}{v^2}$
4. Expand the numerator by distributing the terms
5. Simplify by combining like terms in the numerator
6. Factor the numerator and cancel common factors with the denominator to obtain the final result