Find frac{d y}{d t} at x=3 and y=x^{2}+2 if frac{d x}{d t}=5.

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Step:1 ：We are given the function (y = x^{2} + 2) and we are asked to find (frac{dy}{dt}) at (x = 3) given that (frac{dx}{dt} = 5).Step:2 ：We know that (frac{dy}{dt} = frac{dy}{dx} cdot frac{dx}{dt}) by the chain rule.Step:3 ：So, we first need to find (frac{dy}{dx}), which is the derivative of (y) with respect to (x).Step:4 ：(frac{dy}{dx} = 2x)Step:5 ：Then we can substitute (x = 3) and (frac{dx}{dt} = 5) into the equation to find (frac{dy}{dt}).Step:6 ：(frac{dy}{dt} = 2x cdot frac{dx}{dt} = 2 cdot 3 cdot 5 = 30)Step:7 ：Final Answer: (boxed{30})

Answer

Step: 1 ：We are given the function (y = x^{2} + 2) and we are asked to find (frac{dy}{dt}) at (x = 3) given that (frac{dx}{dt} = 5).Step: 2 ：We know that (frac{dy}{dt} = frac{dy}{dx} cdot frac{dx}{dt}) by the chain rule.Step: 3 ：So, we first need to find (frac{dy}{dx}), which is the derivative of (y) with respect to (x).Step: 4 ：(frac{dy}{dx} = 2x)Step: 5 ：Then we can substitute (x = 3) and (frac{dx}{dt} = 5) into the equation to find (frac{dy}{dt}).Step: 6 ：(frac{dy}{dt} = 2x cdot frac{dx}{dt} = 2 cdot 3 cdot 5 = 30)Step: 7 ：Final Answer: (boxed{30})

Find $\frac{d y}{d t}$ at $x=3$ and $y=x^{2}+2$ if $\frac{d x}{d t}=5$.

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