Acceleration = d/dx (1/2 v^2)

Acceleration = \frac{d}{dx}(\frac{1}{2}v^2):

\displaystyle \boxed{a=\frac{d^2x}{dt^2}=\frac{dv}{dt}=v\frac{dv}{dx}=\frac{d}{dx}(\frac{1}{2}v^2)},

where x is displacement, v is velocity, and a is acceleration.

Proof:
\begin{aligned} a&=\frac{dv}{dt} \text{(this is definition of acceleration)}\\ &=\frac{dv}{dx}\cdot\frac{dx}{dt} \text{(chain rule)}\\ &=v\frac{dv}{dx}. \end{aligned}

Also,
\begin{aligned} \frac{d}{dx}(\frac{1}{2}v^2)&=v\frac{dv}{dx}. \text{(chain rule)} \end{aligned}

Therefore, a=\frac{d}{dx}(\frac{1}{2}v^2).

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Author: mathtuition88

Math and Education Blog

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