Math / Calculus

QuestionDifferentiate both sides of equation $x=x_{0}+v_{0} t+\frac{1}{2} a t^{2}$

Studdy Solution

STEP 1

1. $x$ , $x_0$ , $v_0$ , $t$ , and $a$ are functions of $t$ .
2. We need to find the derivative with respect to $t$ of both sides of the equation $x = x_0 + v_0 t + \frac{1}{2} a t^2$ .

STEP 2

1. Differentiate the left-hand side of the equation with respect to $t$ .
2. Differentiate the right-hand side of the equation with respect to $t$ .
3. Combine and simplify the derivatives to find the final expression.

STEP 3

Differentiate the left-hand side of the equation $x$ with respect to $t$ .
$\frac{d}{dt}(x)$

STEP 4

The derivative of $x$ with respect to $t$ is:
$\frac{dx}{dt}$

STEP 5

Differentiate the first term on the right-hand side, $x_0$ , with respect to $t$ .
$\frac{d}{dt}(x_0)$

STEP 6

Since $x_0$ is a constant, its derivative with respect to $t$ is:
$0$

STEP 7

Differentiate the second term on the right-hand side, $v_0 t$ , with respect to $t$ .
$\frac{d}{dt}(v_0 t)$

STEP 8

Assuming $v_0$ is a constant, the derivative of $v_0 t$ with respect to $t$ is:
$v_0$

STEP 9

Differentiate the third term on the right-hand side, $\frac{1}{2} a t^2$ , with respect to $t$ .
$\frac{d}{dt}\left(\frac{1}{2} a t^2\right)$

STEP 10

Using the chain rule, the derivative of $\frac{1}{2} a t^2$ is:
$\frac{1}{2} a \cdot 2 t = a t$

STEP 11

Combine the derivatives of all terms on the right-hand side:
$0 + v_0 + a t$

STEP 12

Simplify the derivatives to obtain the final expression:
$v_0 + a t$

STEP 13

Equate the derivative of the left-hand side to the derivative of the right-hand side:
$\frac{dx}{dt} = v_0 + a t$
Therefore, the derivative of both sides of the equation $x = x_0 + v_0 t + \frac{1}{2} a t^2$ with respect to $t$ is:
$\frac{dx}{dt} = v_0 + a t$

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