The differentiable functions x and y are related by the following equation: y=sqrtx Also, (dx)/(dt)=12. Find (dy)/(dt) when x=9.

Given function and rate: We are given the function y = sqrt(x) and the rate of change of x with respect to t, which is (dx)/(dt) = 12. We need to find the rate of change of y with respect to t, which is (dy)/(dt), when x = 9.Apply chain rule: To find (dy)/(dt), we can use the chain rule from calculus, which states that (dy)/(dt) = (dy)/(dx) * (dx)/(dt). We already know (dx)/(dt) = 12, so we need to find (dy)/(dx) when x = 9.Find derivative of y: To find (dy)/(dx), we differentiate y = sqrt(x) with respect to x. The derivative of sqrt(x) with respect to x is (1/2)x^(-1/2).Substitute x=9: Now we substitute x = 9 into the derivative (dy)/(dx) to find its value at that point. (dy)/(dx) = (1/2)(9)^(-1/2) = (1/2)(1/sqrt(9)) = (1/2)(1/3) = 1/6.Calculate dy/dt: Now that we have (dy)/(dx) = 1/6 and (dx)/(dt) = 12, we can use the chain rule to find (dy)/(dt). (dy)/(dt) = (dy)/(dx) * (dx)/(dt) = (1/6) * 12 = 2.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The differentiable functions
x and
y are related by the following equation:

y=sqrtx
Also,
(dx)/(dt)=12.
Find
(dy)/(dt) when
x=9.

The differentiable functions $x$ and $y$ are related by the following equation: $\newline$ $y=\sqrt{x}$ $\newline$ Also, $\frac{d x}{d t}=12$ . $\newline$ Find $\frac{d y}{d t}$ when $x=9$ .

View step-by-step help

Home

Math Problems

Algebra 1

Domain and range of square root functions: equations

Full solution

Q. The differentiable functions

x

and

y

are related by the following equation:

\newline

y=\sqrt{x}

\newline

Also,

\frac{d x}{d t}=12

\newline

Find

\frac{d y}{d t}

when

x=9

Given function and rate: We are given the function $y = \sqrt{x}$ and the rate of change of $x$ with respect to $t$ , which is $\frac{dx}{dt} = 12$ . We need to find the rate of change of $y$ with respect to $t$ , which is $\frac{dy}{dt}$ , when $x = 9$ .
Apply chain rule: To find $\frac{dy}{dt}$ , we can use the chain rule from calculus, which states that $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$ . We already know $\frac{dx}{dt} = 12$ , so we need to find $\frac{dy}{dx}$ when $x = 9$ .
Find derivative of $y$ : To find $\frac{dy}{dx}$ , we differentiate $y = \sqrt{x}$ with respect to $x$ . The derivative of $\sqrt{x}$ with respect to $x$ is $\frac{1}{2}x^{-\frac{1}{2}}$ .
Substitute $x=9$ : Now we substitute $x = 9$ into the derivative $\frac{dy}{dx}$ to find its value at that point. $\frac{dy}{dx} = \frac{1}{2}(9)^{-\frac{1}{2}} = \frac{1}{2}(\frac{1}{\sqrt{9}}) = \frac{1}{2}(\frac{1}{3}) = \frac{1}{6}$ .
Calculate $\frac{dy}{dt}$ : Now that we have $\frac{dy}{dx} = \frac{1}{6}$ and $\frac{dx}{dt} = 12$ , we can use the chain rule to find $\frac{dy}{dt}$ . $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} = \frac{1}{6} \times 12 = 2$ .