Questions: Find the second derivative of v with respect to t, where v = 3t^2 + 6t + 13.

Transcript text: Find $\frac{d^{2} v}{d t^{2}}$ \[ v=3 t^{2}+6 t+13 \] \[ \frac{d^{2} v}{d t^{2}}= \]

Solution

Solution Steps

Step 1: Find the first derivative of $ v $ with respect to $ t $

Given the function: \[ v = 3t^{2} + 6t + 13 \] Differentiate $ v $ with respect to $ t $: \[ \frac{dv}{dt} = \frac{d}{dt}(3t^{2}) + \frac{d}{dt}(6t) + \frac{d}{dt}(13) \] \[ \frac{dv}{dt} = 6t + 6 + 0 \] \[ \frac{dv}{dt} = 6t + 6 \]

Step 2: Find the second derivative of $ v $ with respect to $ t $

Differentiate $ \frac{dv}{dt} $ with respect to $ t $: \[ \frac{d^{2}v}{dt^{2}} = \frac{d}{dt}(6t + 6) \] \[ \frac{d^{2}v}{dt^{2}} = 6 + 0 \] \[ \frac{d^{2}v}{dt^{2}} = 6 \]

Step 3: Write the final result

The second derivative of $ v $ with respect to $ t $ is: \[ \frac{d^{2}v}{dt^{2}} = 6 \]