A particle moves along the curve y=7x2+8 in such a way that its x-coordinate is changing at a rate of 7 centimeters per second. At what rate is the particle's y-coordinate changing when the particle is at the point where x=−1x=−1?

We know that the curve is described by [tex]y=7x^{2}+8[/tex] and its x-coordinate is changing at a rate of 7 centimeters per second [tex]\frac{dx}{dt}=7 \:\frac{cm}{s}[/tex]

and we want to find the rate of change of the particle's y-coordinate ([tex]\frac{dy}{dt}[/tex]) when the particles is at the point x = −1.

First we need to find the derivative with respect of time of the curve [tex]y=7x^{2}+8[/tex], we use implicit differentiation.

[tex]\frac{d}{dt} y=\frac{d}{dt}(7x^{2}+8)\\\\\frac{dy}{dt}=\frac{d}{dt}(7x^2)+\frac{d}{dt}(8)\\\\\frac{dy}{dt}=\frac{d}{dt}(7)x^2+7\cdot 2x\frac{dx}{dt} +0\\\\\frac{dy}{dt}=0\cdot x^2+14x\frac{dx}{dt}+0\\\\\frac{dy}{dt}=14x\frac{dx}{dt}[/tex]

Now, we can substitute the values that we know

[tex]\frac{dx}{dt}=7 \:\frac{cm}{s}[/tex]

x = −1

[tex]\frac{dy}{dt}=14x\frac{dx}{dt}\\\\\frac{dy}{dt}=14\cdot (-1)\cdot 7\\\\\frac{dy}{dt}=-98 \:\frac{cm}{s}[/tex]

The rate of change of the particle's y-coordinate when the particles is at the point x = −1 is [tex]\frac{dy}{dt}=-98 \:\frac{cm}{s}[/tex]