Calculus Examples

$y = (z^{2} + e^{z}) \sqrt{z}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z}$ as $z^{\frac{1}{2}}$ .

$\frac{d}{d z} [(z^{2} + e^{z}) z^{\frac{1}{2}}]$

Step 2

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 $f (z) = z^{2} + e^{z}$ and $g (z) = z^{\frac{1}{2}}$ .

$(z^{2} + e^{z}) \frac{d}{d z} [z^{\frac{1}{2}}] + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 3

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

$(z^{2} + e^{z}) (\frac{1}{2} z^{\frac{1}{2} - 1}) + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(z^{2} + e^{z}) (\frac{1}{2} z^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 5

Combine $- 1$ and $\frac{2}{2}$ .

$(z^{2} + e^{z}) (\frac{1}{2} z^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 6

Combine the numerators over the common denominator.

$(z^{2} + e^{z}) (\frac{1}{2} z^{\frac{1 - 1 \cdot 2}{2}}) + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$(z^{2} + e^{z}) (\frac{1}{2} z^{\frac{1 - 2}{2}}) + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 7.2

Subtract $2$ from $1$ .

$(z^{2} + e^{z}) (\frac{1}{2} z^{\frac{- 1}{2}}) + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 8

Differentiate.

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Step 8.1

Move the negative in front of the fraction.

$(z^{2} + e^{z}) (\frac{1}{2} z^{- \frac{1}{2}}) + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 8.2

Combine fractions.

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Step 8.2.1

Combine $\frac{1}{2}$ and $z^{- \frac{1}{2}}$ .

$(z^{2} + e^{z}) \frac{z^{- \frac{1}{2}}}{2} + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 8.2.2

Move $z^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(z^{2} + e^{z}) \frac{1}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} \frac{d}{d z} [z^{2} + e^{z}]$

Step 8.3

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

$(z^{2} + e^{z}) \frac{1}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} (\frac{d}{d z} [z^{2}] + \frac{d}{d z} [e^{z}])$

Step 8.4

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

$(z^{2} + e^{z}) \frac{1}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} (2 z + \frac{d}{d z} [e^{z}])$

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Apply the distributive property.

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

Step 10.3

Combine terms.

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Step 10.3.1

Combine $z^{2}$ and $\frac{1}{2 z^{\frac{1}{2}}}$ .

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

Step 10.3.2

Move $z^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 10.3.3

Multiply $z^{2}$ by $z^{- \frac{1}{2}}$ by adding the exponents.

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Step 10.3.3.1

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

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

Step 10.3.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 10.3.3.3

Combine $2$ and $\frac{2}{2}$ .

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

Step 10.3.3.4

Combine the numerators over the common denominator.

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

Step 10.3.3.5

Simplify the numerator.

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Step 10.3.3.5.1

Multiply $2$ by $2$ .

$\frac{z^{\frac{4 - 1}{2}}}{2} + e^{z} \frac{1}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} (2 z) + z^{\frac{1}{2}} e^{z}$

Step 10.3.3.5.2

Subtract $1$ from $4$ .

$\frac{z^{\frac{3}{2}}}{2} + e^{z} \frac{1}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} (2 z) + z^{\frac{1}{2}} e^{z}$

Step 10.3.4

Combine $e^{z}$ and $\frac{1}{2 z^{\frac{1}{2}}}$ .

$\frac{z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} (2 z) + z^{\frac{1}{2}} e^{z}$

Step 10.3.5

Multiply $z^{\frac{1}{2}}$ by $z$ by adding the exponents.

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Step 10.3.5.1

Move $z$ .

$\frac{z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z \cdot z^{\frac{1}{2}} \cdot 2 + z^{\frac{1}{2}} e^{z}$

Step 10.3.5.2

Multiply $z$ by $z^{\frac{1}{2}}$ .

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Step 10.3.5.2.1

Raise $z$ to the power of $1$ .

$\frac{z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{1} z^{\frac{1}{2}} \cdot 2 + z^{\frac{1}{2}} e^{z}$

Step 10.3.5.2.2

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

$\frac{z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{1 + \frac{1}{2}} \cdot 2 + z^{\frac{1}{2}} e^{z}$

Step 10.3.5.3

Write $1$ as a fraction with a common denominator.

$\frac{z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{2}{2} + \frac{1}{2}} \cdot 2 + z^{\frac{1}{2}} e^{z}$

Step 10.3.5.4

Combine the numerators over the common denominator.

$\frac{z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{2 + 1}{2}} \cdot 2 + z^{\frac{1}{2}} e^{z}$

Step 10.3.5.5

Add $2$ and $1$ .

$\frac{z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{3}{2}} \cdot 2 + z^{\frac{1}{2}} e^{z}$

Step 10.3.6

Move $2$ to the left of $z^{\frac{3}{2}}$ .

$\frac{z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + 2 \cdot z^{\frac{3}{2}} + z^{\frac{1}{2}} e^{z}$

Step 10.3.7

To write $2 z^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{z^{\frac{3}{2}}}{2} + 2 z^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} e^{z}$

Step 10.3.8

Combine $2 z^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$\frac{z^{\frac{3}{2}}}{2} + \frac{2 z^{\frac{3}{2}} \cdot 2}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} e^{z}$

Step 10.3.9

Combine the numerators over the common denominator.

$\frac{z^{\frac{3}{2}} + 2 z^{\frac{3}{2}} \cdot 2}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} e^{z}$

Step 10.3.10

Multiply $2$ by $2$ .

$\frac{z^{\frac{3}{2}} + 4 z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} e^{z}$

Step 10.3.11

Add $z^{\frac{3}{2}}$ and $4 z^{\frac{3}{2}}$ .

$\frac{5 z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}} + z^{\frac{1}{2}} e^{z}$

Step 10.4

Reorder terms.

$e^{z} z^{\frac{1}{2}} + \frac{5 z^{\frac{3}{2}}}{2} + \frac{e^{z}}{2 z^{\frac{1}{2}}}$