QuestionJune 18, 2025

Use the quotient rule to find the derivative of the following. y=(x^2-2x+5)/(x^2)+7 (dy)/(dx)=square

Use the quotient rule to find the derivative of the following. y=(x^2-2x+5)/(x^2)+7 (dy)/(dx)=square
Use the quotient rule to find the derivative of the following. y=(x^2-2x+5)/(x^2)+7 (dy)/(dx)=square

Solution
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Answer

\frac{dy}{dx} = \frac{2x^2 + 4x - 14}{(x^2 + 7)^2} Explanation 1. Identify the functions Let u = x^2 - 2x + 5 and v = x^2 + 7. 2. Apply the Quotient Rule The quotient rule is \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}. 3. Differentiate u \frac{du}{dx} = 2x - 2. 4. Differentiate v \frac{dv}{dx} = 2x. 5. Substitute into the Quotient Rule \frac{dy}{dx} = \frac{(x^2 + 7)(2x - 2) - (x^2 - 2x + 5)(2x)}{(x^2 + 7)^2}. 6. Simplify the expression Expand and simplify the numerator: (2x^3 - 2x^2 + 14x - 14) - (2x^3 - 4x^2 + 10x). Combine like terms: 2x^3 - 2x^2 + 14x - 14 - 2x^3 + 4x^2 - 10x = 2x^2 + 4x - 14.

Explanation

1. Identify the functions<br /> Let $u = x^2 - 2x + 5$ and $v = x^2 + 7$.<br />2. Apply the Quotient Rule<br /> The quotient rule is $\frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}$.<br />3. Differentiate $u$<br /> $\frac{du}{dx} = 2x - 2$.<br />4. Differentiate $v$<br /> $\frac{dv}{dx} = 2x$.<br />5. Substitute into the Quotient Rule<br /> $\frac{dy}{dx} = \frac{(x^2 + 7)(2x - 2) - (x^2 - 2x + 5)(2x)}{(x^2 + 7)^2}$.<br />6. Simplify the expression<br /> Expand and simplify the numerator: $(2x^3 - 2x^2 + 14x - 14) - (2x^3 - 4x^2 + 10x)$.<br /> Combine like terms: $2x^3 - 2x^2 + 14x - 14 - 2x^3 + 4x^2 - 10x = 2x^2 + 4x - 14$.
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