Munoz Whittaker
12/17/2023 · Elementary School
7. Find \( y^{\prime} \) if \( y=\frac{x}{x+y} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( y=\frac{x}{x+y} \).
Find the first derivative by following steps:
- step0: Find the derivative with respect to \(x\):
\(y=\frac{x}{\left(x+y\right)}\)
- step1: Simplify the expression:
\(y=\frac{x}{x+y}\)
- step2: Take the derivative:
\(\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(\frac{x}{x+y}\right)\)
- step3: Calculate the derivative:
\(\frac{dy}{dx}=\frac{d}{dx}\left(\frac{x}{x+y}\right)\)
- step4: Calculate the derivative:
\(\frac{dy}{dx}=\frac{y-x\frac{dy}{dx}}{\left(x+y\right)^{2}}\)
- step5: Cross multiply:
\(\frac{dy}{dx}\times \left(x+y\right)^{2}=y-x\frac{dy}{dx}\)
- step6: Simplify the equation:
\(\left(x+y\right)^{2}\frac{dy}{dx}=y-x\frac{dy}{dx}\)
- step7: Move the variable to the left side:
\(\left(x+y\right)^{2}\frac{dy}{dx}+x\frac{dy}{dx}=y\)
- step8: Collect like terms:
\(\left(\left(x+y\right)^{2}+x\right)\frac{dy}{dx}=y\)
- step9: Divide both sides:
\(\frac{\left(\left(x+y\right)^{2}+x\right)\frac{dy}{dx}}{\left(x+y\right)^{2}+x}=\frac{y}{\left(x+y\right)^{2}+x}\)
- step10: Divide the numbers:
\(\frac{dy}{dx}=\frac{y}{\left(x+y\right)^{2}+x}\)
- step11: Calculate:
\(\frac{dy}{dx}=\frac{y}{x^{2}+2xy+y^{2}+x}\)
The derivative of \( y \) with respect to \( x \) is given by:
\[ \frac{dy}{dx} = \frac{y}{x^{2}+2xy+y^{2}+x} \]
Quick Answer
\[ \frac{dy}{dx} = \frac{y}{x^{2}+2xy+y^{2}+x} \]
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