Question

Find dy/dt at x = -2 if y = -2x^2 - 5 and dx/dt = -5 dy/dt = ???

Answer 1

$\frac{dy}{dt}=-40$

Solution:

Given data:

$y=-2 x^{2}-5$ and $\frac{dx}{dt}=-5$

To find $\frac{dy}{dt}$:

$y=-2 x^{2}-5$

Differentiate y with respect to t.

$\frac{dy}{dt}=\frac{d}{dt}(-2x^2-5)$

$\frac{dy}{dt}=\frac{d}{dt}(-2x^2)-\frac{d}{dt}(5)$

Apply the differentiation rule: $\frac{d}{d x}\left(x^{n}\right)=n \cdot x^{n-1}$

$\frac{dy}{dt}=2(-2x^{2-1})\cdot \frac{dx}{dt} -\frac{d}{dt}(5)$

$\frac{dy}{dt}=-4x\cdot \frac{dx}{dt} -\frac{d}{dt}(5)$

Apply the differentiation rule: $\frac{d}{dt}a=0$

$\frac{dy}{dt}=-4x\cdot \frac{dx}{dt} -0$

$\frac{dy}{dt}=-4x\cdot \frac{dx}{dt}$

At x = –2 and $\frac{dx}{dt}=-5$

$\frac{dy}{dt}=-4(-2)\cdot (-5)$

$\frac{dy}{dt}=-40$

Therefore, $\frac{dy}{dt}=-40$.