Assume that x and y are both differential y=sqrtx (a) Find (dy)/(dt), given x=16 and (dx)/(dt)=6.

Given function and task: We are given the function y = sqrt(x) and we need to find (dy)/(dt) when x = 16 and (dx)/(dt) = 6. To find (dy)/(dt), we will first find the derivative of y with respect to x, which is (dy)/(dx), and then use the chain rule to find (dy)/(dt).Derivative of y with respect to x: The derivative of y with respect to x, when y = sqrt(x), is (dy)/(dx) = 1/(2sqrt(x)). This is because the derivative of sqrt(x) with respect to x is 1/(2sqrt(x)).Applying chain rule: Now we apply the chain rule to find (dy)/(dt). The chain rule states that (dy)/(dt) = (dy)/(dx) * (dx)/(dt). We already have (dx)/(dt) = 6, and we need to evaluate (dy)/(dx) at x = 16.Substitute x = 16: Substitute x = 16 into the derivative (dy)/(dx) to get (dy)/(dx) = 1/(2sqrt(16)) = 1/(2*4) = 1/8.Finding (dy)/(dt): Now we can find (dy)/(dt) by multiplying (dy)/(dx) by (dx)/(dt). So, (dy)/(dt) = (1/8) * 6 = 6/8 = 3/4.

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Assume that $x$ and $y$ are both differential $\newline$ $y=\sqrt{x}$ $\newline$ (a) Find $\frac{dy}{dt}$ , given $x=16$ and $\frac{dx}{dt}=6$ .

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Algebra 1

Domain and range of square root functions: equations

Full solution

Q. Assume that

x

and

y

are both differential

\newline

y=\sqrt{x}

\newline

(a) Find

\frac{dy}{dt}

, given

x=16

and

\frac{dx}{dt}=6

Given function and task: We are given the function $y = \sqrt{x}$ and we need to find $\frac{dy}{dt}$ when $x = 16$ and $\frac{dx}{dt} = 6$ . To find $\frac{dy}{dt}$ , we will first find the derivative of $y$ with respect to $x$ , which is $\frac{dy}{dx}$ , and then use the chain rule to find $\frac{dy}{dt}$ .
Derivative of $y$ with respect to $x$ : The derivative of $y$ with respect to $x$ , when $y = \sqrt{x}$ , is $\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$ . This is because the derivative of $\sqrt{x}$ with respect to $x$ is $\frac{1}{2\sqrt{x}}$ .
Applying chain rule: Now we apply the chain rule to find $\frac{dy}{dt}$ . The chain rule states that $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$ . We already have $\frac{dx}{dt} = 6$ , and we need to evaluate $\frac{dy}{dx}$ at $x = 16$ .
Substitute $x = 16$ : Substitute $x = 16$ into the derivative $\frac{dy}{dx}$ to get $\frac{dy}{dx} = \frac{1}{2\sqrt{16}} = \frac{1}{2\cdot 4} = \frac{1}{8}$ .
Finding $(dy)/(dt)$ : Now we can find $(dy)/(dt)$ by multiplying $(dy)/(dx)$ by $(dx)/(dt)$ . So, $(dy)/(dt) = (1/8) \times 6 = 6/8 = 3/4$ .