Question

The velocity of a moving vehicle is given by the equation $𝑣 = (2𝑡 + 3)^4$. Use the Chain Rule to determine an equation for the acceleration when 𝑎 = 𝑑𝑣 𝑑𝑡 .

Answer 1

Given: $v=(2t+3)^4$

Require to find $a=\frac{dv}{dt}$ using Chain Rule.

Recollect the following (Chain Rule):

If $y=f(u)$ and $u=g(x)$ , then $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$

Let $u=(2t+3)$

Then $v=u^4,u=2t+3$

$\Rightarrow \frac{dv}{du}=4u^3,\frac{du}{dt}=2(1)+0=2$

Using Chain Rule, we get

$\frac{dv}{dt}=4u^3(2)=8u^3$

Substituting $u=2t+3,$ we get

$\frac{dv}{dt}=8(2t+3)^3$

Therefore the acceleration $a=\frac{dv}{dt}=8(2t+3)^3$