Calculus Examples

$s = t^{2} - t$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (t^{2} - t)$

Step 2

The derivative of $s$ with respect to $t$ is $s'$ .

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $t^{2} - t$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- t]$ .

$\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- t]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$2 t + \frac{d}{d t} [- t]$

Step 3.2

Evaluate $\frac{d}{d t} [- t]$ .

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Step 3.2.1

Since $- 1$ is constant with respect to $t$ , the derivative of $- t$ with respect to $t$ is $- \frac{d}{d t} [t]$ .

$2 t - \frac{d}{d t} [t]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 t - 1 \cdot 1$

Step 3.2.3

Multiply $- 1$ by $1$ .

$2 t - 1$

Step 4

Reform the equation by setting the left side equal to the right side.

$s' = 2 t - 1$

Step 5

Replace $s'$ with $\frac{d s}{d t}$ .

$\frac{d s}{d t} = 2 t - 1$