Calculus Examples

$\ln^{7} (z) + \ln (z^{7})$

Step 1

By the Sum Rule, the derivative of $\ln^{7} (z) + \ln (z^{7})$ with respect to $z$ is $\frac{d}{d z} [\ln^{7} (z)] + \frac{d}{d z} [\ln (z^{7})]$ .

$\frac{d}{d z} [\ln^{7} (z)] + \frac{d}{d z} [\ln (z^{7})]$

Step 2

Evaluate $\frac{d}{d z} [\ln^{7} (z)]$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = z^{7}$ and $g (z) = \ln (z)$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $\ln (z)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{7}] \frac{d}{d z} [\ln (z)] + \frac{d}{d z} [\ln (z^{7})]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 7$ .

$7 {u_{1}}^{6} \frac{d}{d z} [\ln (z)] + \frac{d}{d z} [\ln (z^{7})]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $\ln (z)$ .

$7 \ln^{6} (z) \frac{d}{d z} [\ln (z)] + \frac{d}{d z} [\ln (z^{7})]$

Step 2.2

The derivative of $\ln (z)$ with respect to $z$ is $\frac{1}{z}$ .

$7 \ln^{6} (z) \frac{1}{z} + \frac{d}{d z} [\ln (z^{7})]$

Step 2.3

Combine $\frac{1}{z}$ and $7$ .

$\frac{7}{z} \ln^{6} (z) + \frac{d}{d z} [\ln (z^{7})]$

Step 2.4

Combine $\frac{7}{z}$ and $\ln^{6} (z)$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{d}{d z} [\ln (z^{7})]$

Step 3

Evaluate $\frac{d}{d z} [\ln (z^{7})]$ .

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = \ln (z)$ and $g (z) = z^{7}$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{2}$ as $z^{7}$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d z} [z^{7}]$

Step 3.1.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{1}{u_{2}} \frac{d}{d z} [z^{7}]$

Step 3.1.3

Replace all occurrences of $u_{2}$ with $z^{7}$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{1}{z^{7}} \frac{d}{d z} [z^{7}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d z} [z^{n}]$ is $n z^{n - 1}$ where $n = 7$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{1}{z^{7}} (7 z^{6})$

Step 3.3

Combine $7$ and $\frac{1}{z^{7}}$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{7}{z^{7}} z^{6}$

Step 3.4

Combine $\frac{7}{z^{7}}$ and $z^{6}$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{7 z^{6}}{z^{7}}$

Step 3.5

Cancel the common factor of $z^{6}$ and $z^{7}$ .

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Step 3.5.1

Factor $z^{6}$ out of $7 z^{6}$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{z^{6} \cdot 7}{z^{7}}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $z^{6}$ out of $z^{7}$ .

$\frac{7 \ln^{6} (z)}{z} + \frac{z^{6} \cdot 7}{z^{6} z}$

Step 3.5.2.2

Cancel the common factor.

$\frac{7 \ln^{6} (z)}{z} + \frac{z^{6} \cdot 7}{z^{6} z}$

Step 3.5.2.3

Rewrite the expression.

$\frac{7 \ln^{6} (z)}{z} + \frac{7}{z}$