Calculus Examples

$(1 - e^{z}) (z + e^{z})$

Step 1

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

$(1 - e^{z}) \frac{d}{d z} [z + e^{z}] + (z + e^{z}) \frac{d}{d z} [1 - e^{z}]$

Step 2

Differentiate.

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Step 2.1

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

$(1 - e^{z}) (\frac{d}{d z} [z] + \frac{d}{d z} [e^{z}]) + (z + e^{z}) \frac{d}{d z} [1 - e^{z}]$

Step 2.2

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

$(1 - e^{z}) (1 + \frac{d}{d z} [e^{z}]) + (z + e^{z}) \frac{d}{d z} [1 - e^{z}]$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d z} [a^{z}]$ is $a^{z} \ln (a)$ where $a$ = $e$ .

$(1 - e^{z}) (1 + e^{z}) + (z + e^{z}) \frac{d}{d z} [1 - e^{z}]$

Step 4

Differentiate.

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Step 4.1

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

$(1 - e^{z}) (1 + e^{z}) + (z + e^{z}) (\frac{d}{d z} [1] + \frac{d}{d z} [- e^{z}])$

Step 4.2

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

$(1 - e^{z}) (1 + e^{z}) + (z + e^{z}) (0 + \frac{d}{d z} [- e^{z}])$

Step 4.3

Add $0$ and $\frac{d}{d z} [- e^{z}]$ .

$(1 - e^{z}) (1 + e^{z}) + (z + e^{z}) \frac{d}{d z} [- e^{z}]$

Step 4.4

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

$(1 - e^{z}) (1 + e^{z}) + (z + e^{z}) (- \frac{d}{d z} [e^{z}])$

Step 5

Differentiate using the Exponential Rule which states that $\frac{d}{d z} [a^{z}]$ is $a^{z} \ln (a)$ where $a$ = $e$ .

$(1 - e^{z}) (1 + e^{z}) + (z + e^{z}) (- e^{z})$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$(1 - e^{z}) (1 + e^{z}) + z (- e^{z}) + e^{z} (- e^{z})$

Step 6.2

Combine terms.

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Step 6.2.1

Multiply $e^{z}$ by $e^{z}$ by adding the exponents.

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Step 6.2.1.1

Move $e^{z}$ .

$(1 - e^{z}) (1 + e^{z}) + z (- e^{z}) + e^{z} e^{z} \cdot - 1$

Step 6.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(1 - e^{z}) (1 + e^{z}) + z (- e^{z}) + e^{z + z} \cdot - 1$

Step 6.2.1.3

Add $z$ and $z$ .

$(1 - e^{z}) (1 + e^{z}) + z (- e^{z}) + e^{2 z} \cdot - 1$

Step 6.2.2

Move $- 1$ to the left of $e^{2 z}$ .

$(1 - e^{z}) (1 + e^{z}) + z (- e^{z}) - 1 \cdot e^{2 z}$

Step 6.2.3

Rewrite $- 1 e^{2 z}$ as $- e^{2 z}$ .

$(1 - e^{z}) (1 + e^{z}) + z (- e^{z}) - e^{2 z}$

Step 6.3

Reorder terms.

$(1 + e^{z}) (1 - e^{z}) - e^{z} z - e^{2 z}$

Step 6.4

Simplify each term.

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Step 6.4.1

Expand $(1 + e^{z}) (1 - e^{z})$ using the FOIL Method.

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Step 6.4.1.1

Apply the distributive property.

$1 (1 - e^{z}) + e^{z} (1 - e^{z}) - e^{z} z - e^{2 z}$

Step 6.4.1.2

Apply the distributive property.

$1 \cdot 1 + 1 (- e^{z}) + e^{z} (1 - e^{z}) - e^{z} z - e^{2 z}$

Step 6.4.1.3

Apply the distributive property.

$1 \cdot 1 + 1 (- e^{z}) + e^{z} \cdot 1 + e^{z} (- e^{z}) - e^{z} z - e^{2 z}$

Step 6.4.2

Simplify and combine like terms.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Multiply $1$ by $1$ .

$1 + 1 (- e^{z}) + e^{z} \cdot 1 + e^{z} (- e^{z}) - e^{z} z - e^{2 z}$

Step 6.4.2.1.2

Multiply $- e^{z}$ by $1$ .

$1 - e^{z} + e^{z} \cdot 1 + e^{z} (- e^{z}) - e^{z} z - e^{2 z}$

Step 6.4.2.1.3

Multiply $e^{z}$ by $1$ .

$1 - e^{z} + e^{z} + e^{z} (- e^{z}) - e^{z} z - e^{2 z}$

Step 6.4.2.1.4

Rewrite using the commutative property of multiplication.

$1 - e^{z} + e^{z} - e^{z} e^{z} - e^{z} z - e^{2 z}$

Step 6.4.2.1.5

Multiply $e^{z}$ by $e^{z}$ by adding the exponents.

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Step 6.4.2.1.5.1

Move $e^{z}$ .

$1 - e^{z} + e^{z} - (e^{z} e^{z}) - e^{z} z - e^{2 z}$

Step 6.4.2.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 - e^{z} + e^{z} - e^{z + z} - e^{z} z - e^{2 z}$

Step 6.4.2.1.5.3

Add $z$ and $z$ .

$1 - e^{z} + e^{z} - e^{2 z} - e^{z} z - e^{2 z}$

Step 6.4.2.2

Add $- e^{z}$ and $e^{z}$ .

$1 + 0 - e^{2 z} - e^{z} z - e^{2 z}$

Step 6.4.2.3

Add $1$ and $0$ .

$1 - e^{2 z} - e^{z} z - e^{2 z}$

Step 6.5

Subtract $e^{2 z}$ from $- e^{2 z}$ .

$1 - e^{z} z - 2 e^{2 z}$

Step 6.6

Reorder factors in $1 - e^{z} z - 2 e^{2 z}$ .

$1 - z e^{z} - 2 e^{2 z}$