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Let xx and yy be functions of tt with y=x2+x12y = x^2 + x^{\frac{1}{2}}. If dxdt=18\frac{dx}{dt} = \frac{1}{8}, what is dydt\frac{dy}{dt} when x=4x = 4?\newlineWrite an exact, simplified answer.

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Write and solve direct variation equationsright chevron icon

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Q. Let xx and yy be functions of tt with y=x2+x12y = x^2 + x^{\frac{1}{2}}. If dxdt=18\frac{dx}{dt} = \frac{1}{8}, what is dydt\frac{dy}{dt} when x=4x = 4?\newlineWrite an exact, simplified answer.
  1. Identify Relationship: Identify the relationship and differentiate.\newlineGiven y=x2+x1/2y = x^2 + x^{1/2}, we need to find dydt\frac{dy}{dt} using the chain rule.\newlinedydx=2x+(12)x−1/2\frac{dy}{dx} = 2x + \left(\frac{1}{2}\right)x^{-1/2}
  2. Substitute dxdt\frac{dx}{dt}: Substitute dxdt\frac{dx}{dt} into the equation.\newlineGiven dxdt=18\frac{dx}{dt} = \frac{1}{8}, substitute and find dydt\frac{dy}{dt}.\newlinedydt=dydx⋅dxdt=(2x+(12)x(−12))⋅(18)\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} = (2x + (\frac{1}{2})x^{(-\frac{1}{2})}) \cdot (\frac{1}{8})
  3. Evaluate at x=4x=4: Evaluate dydt\frac{dy}{dt} at x=4x = 4.\newlineSubstitute x=4x = 4 into dydt\frac{dy}{dt}.\newline\frac{dy}{dt} = (\(2\cdot44 + (\frac{11}{22})\cdot44^{(-\frac{11}{22})}) \cdot (\frac{11}{88})\newline = (88 + (\frac{11}{22})\cdot\frac{11}{22}) \cdot (\frac{11}{88})\newline = (88 + \frac{11}{44}) \cdot (\frac{11}{88})\newline = (\frac{3333}{44}) \cdot (\frac{11}{88})\newline = \frac{3333}{3232}

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