Calculus Examples

$z \ln (x^{2} y \cos (z))$

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 $f (z) = z$ and $g (z) = \ln (x^{2} y \cos (z))$ .

$z \frac{d}{d z} [\ln (x^{2} y \cos (z))] + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 2

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 $g (z) = x^{2} y \cos (z)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2} y \cos (z)$ .

$z (\frac{d}{d u} [\ln (u)] \frac{d}{d z} [x^{2} y \cos (z)]) + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 2.2

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

$z (\frac{1}{u} \frac{d}{d z} [x^{2} y \cos (z)]) + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 2.3

Replace all occurrences of $u$ with $x^{2} y \cos (z)$ .

$z (\frac{1}{x^{2} y \cos (z)} \frac{d}{d z} [x^{2} y \cos (z)]) + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 3

Differentiate using the Constant Multiple Rule.

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Step 3.1

Combine $\frac{1}{x^{2} y \cos (z)}$ and $z$ .

$\frac{z}{x^{2} y \cos (z)} \frac{d}{d z} [x^{2} y \cos (z)] + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 3.2

Since $x^{2} y$ is constant with respect to $z$ , the derivative of $x^{2} y \cos (z)$ with respect to $z$ is $x^{2} y \frac{d}{d z} [\cos (z)]$ .

$\frac{z}{x^{2} y \cos (z)} (x^{2} y \frac{d}{d z} [\cos (z)]) + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $x^{2}$ and $\frac{z}{x^{2} y \cos (z)}$ .

$\frac{x^{2} z}{x^{2} y \cos (z)} (y \frac{d}{d z} [\cos (z)]) + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 3.3.2

Combine $y$ and $\frac{x^{2} z}{x^{2} y \cos (z)}$ .

$\frac{y (x^{2} z)}{x^{2} y \cos (z)} \frac{d}{d z} [\cos (z)] + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 3.3.3

Cancel the common factor of $y$ .

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Step 3.3.3.1

Cancel the common factor.

$\frac{y x^{2} z}{x^{2} y \cos (z)} \frac{d}{d z} [\cos (z)] + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 3.3.3.2

Rewrite the expression.

$\frac{x^{2} z}{x^{2} \cos (z)} \frac{d}{d z} [\cos (z)] + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 3.3.4

Cancel the common factor of $x^{2}$ .

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Step 3.3.4.1

Cancel the common factor.

$\frac{x^{2} z}{x^{2} \cos (z)} \frac{d}{d z} [\cos (z)] + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 3.3.4.2

Rewrite the expression.

$\frac{z}{\cos (z)} \frac{d}{d z} [\cos (z)] + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 4

The derivative of $\cos (z)$ with respect to $z$ is $- \sin (z)$ .

$\frac{z}{\cos (z)} (- \sin (z)) + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 5

Combine $\frac{z}{\cos (z)}$ and $\sin (z)$ .

$- \frac{z \sin (z)}{\cos (z)} + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 6

Convert from $\frac{z \sin (z)}{\cos (z)}$ to $z \tan (z)$ .

$- (z \tan (z)) + \ln (x^{2} y \cos (z)) \frac{d}{d z} [z]$

Step 7

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

$- z \tan (z) + \ln (x^{2} y \cos (z)) \cdot 1$

Step 8

Multiply $\ln (x^{2} y \cos (z))$ by $1$ .

$- z \tan (z) + \ln (x^{2} y \cos (z))$