Calculus Examples

$f (z) = 3 x y^{4} + \ln (4 x z^{2}) - 6 x y z^{2} + 6 y z$

Step 1

By the Sum Rule, the derivative of $3 x y^{4} + \ln (4 x z^{2}) - 6 x y z^{2} + 6 y z$ with respect to $x$ is $\frac{d}{d x} [3 x y^{4}] + \frac{d}{d x} [\ln (4 x z^{2})] + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$ .

$\frac{d}{d x} [3 x y^{4}] + \frac{d}{d x} [\ln (4 x z^{2})] + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 2

Evaluate $\frac{d}{d x} [3 x y^{4}]$ .

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Step 2.1

Since $3 y^{4}$ is constant with respect to $x$ , the derivative of $3 x y^{4}$ with respect to $x$ is $3 y^{4} \frac{d}{d x} [x]$ .

$3 y^{4} \frac{d}{d x} [x] + \frac{d}{d x} [\ln (4 x z^{2})] + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 2.2

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

$3 y^{4} \cdot 1 + \frac{d}{d x} [\ln (4 x z^{2})] + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 2.3

Multiply $3$ by $1$ .

$3 y^{4} + \frac{d}{d x} [\ln (4 x z^{2})] + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3

Evaluate $\frac{d}{d x} [\ln (4 x z^{2})]$ .

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 4 x z^{2}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $4 x z^{2}$ .

$3 y^{4} + \frac{d}{d u} [\ln (u)] \frac{d}{d x} [4 x z^{2}] + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.1.2

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

$3 y^{4} + \frac{1}{u} \frac{d}{d x} [4 x z^{2}] + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.1.3

Replace all occurrences of $u$ with $4 x z^{2}$ .

$3 y^{4} + \frac{1}{4 x z^{2}} \frac{d}{d x} [4 x z^{2}] + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.2

Since $4 z^{2}$ is constant with respect to $x$ , the derivative of $4 x z^{2}$ with respect to $x$ is $4 z^{2} \frac{d}{d x} [x]$ .

$3 y^{4} + \frac{1}{4 x z^{2}} (4 z^{2} \frac{d}{d x} [x]) + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.3

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

$3 y^{4} + \frac{1}{4 x z^{2}} (4 z^{2} \cdot 1) + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.4

Multiply $4$ by $1$ .

$3 y^{4} + \frac{1}{4 x z^{2}} (4 z^{2}) + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.5

Combine $4$ and $\frac{1}{4 x z^{2}}$ .

$3 y^{4} + \frac{4}{4 x z^{2}} z^{2} + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.6

Combine $\frac{4}{4 x z^{2}}$ and $z^{2}$ .

$3 y^{4} + \frac{4 z^{2}}{4 x z^{2}} + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.7

Cancel the common factor of $4$ .

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Step 3.7.1

Cancel the common factor.

$3 y^{4} + \frac{4 z^{2}}{4 x z^{2}} + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.7.2

Rewrite the expression.

$3 y^{4} + \frac{z^{2}}{x z^{2}} + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.8

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

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Step 3.8.1

Cancel the common factor.

$3 y^{4} + \frac{z^{2}}{x z^{2}} + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 3.8.2

Rewrite the expression.

$3 y^{4} + \frac{1}{x} + \frac{d}{d x} [- 6 x y z^{2}] + \frac{d}{d x} [6 y z]$

Step 4

Evaluate $\frac{d}{d x} [- 6 x y z^{2}]$ .

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Step 4.1

Since $- 6 y z^{2}$ is constant with respect to $x$ , the derivative of $- 6 x y z^{2}$ with respect to $x$ is $- 6 y z^{2} \frac{d}{d x} [x]$ .

$3 y^{4} + \frac{1}{x} - 6 y z^{2} \frac{d}{d x} [x] + \frac{d}{d x} [6 y z]$

Step 4.2

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

$3 y^{4} + \frac{1}{x} - 6 y z^{2} \cdot 1 + \frac{d}{d x} [6 y z]$

Step 4.3

Multiply $- 6$ by $1$ .

$3 y^{4} + \frac{1}{x} - 6 y z^{2} + \frac{d}{d x} [6 y z]$

Step 5

Since $6 y z$ is constant with respect to $x$ , the derivative of $6 y z$ with respect to $x$ is $0$ .

$3 y^{4} + \frac{1}{x} - 6 y z^{2} + 0$

Step 6

Simplify.

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Step 6.1

Add $3 y^{4} + \frac{1}{x} - 6 y z^{2}$ and $0$ .

$3 y^{4} + \frac{1}{x} - 6 y z^{2}$

Step 6.2

Reorder terms.

$3 y^{4} - 6 z^{2} y + \frac{1}{x}$