Calculus Examples

$z = (t - 1) (t + 1)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (z) = \frac{d}{d t} ((t - 1) (t + 1))$

Step 2

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

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t - 1$ and $g (t) = t + 1$ .

$(t - 1) \frac{d}{d t} [t + 1] + (t + 1) \frac{d}{d t} [t - 1]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $t + 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [1]$ .

$(t - 1) (\frac{d}{d t} [t] + \frac{d}{d t} [1]) + (t + 1) \frac{d}{d t} [t - 1]$

Step 3.2.2

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

$(t - 1) (1 + \frac{d}{d t} [1]) + (t + 1) \frac{d}{d t} [t - 1]$

Step 3.2.3

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$(t - 1) (1 + 0) + (t + 1) \frac{d}{d t} [t - 1]$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

$(t - 1) \cdot 1 + (t + 1) \frac{d}{d t} [t - 1]$

Step 3.2.4.2

Multiply $t - 1$ by $1$ .

$t - 1 + (t + 1) \frac{d}{d t} [t - 1]$

Step 3.2.5

By the Sum Rule, the derivative of $t - 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 1]$ .

$t - 1 + (t + 1) (\frac{d}{d t} [t] + \frac{d}{d t} [- 1])$

Step 3.2.6

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

$t - 1 + (t + 1) (1 + \frac{d}{d t} [- 1])$

Step 3.2.7

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$t - 1 + (t + 1) (1 + 0)$

Step 3.2.8

Simplify by adding terms.

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Step 3.2.8.1

Add $1$ and $0$ .

$t - 1 + (t + 1) \cdot 1$

Step 3.2.8.2

Multiply $t + 1$ by $1$ .

$t - 1 + t + 1$

Step 3.2.8.3

Add $t$ and $t$ .

$2 t - 1 + 1$

Step 3.2.8.4

Simplify by adding numbers.

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Step 3.2.8.4.1

Add $- 1$ and $1$ .

$2 t + 0$

Step 3.2.8.4.2

Add $2 t$ and $0$ .

$2 t$

Step 4

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

$z' = 2 t$

Step 5

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

$\frac{d z}{d t} = 2 t$