Calculus Examples

$y = x^{4} \ln (x - \frac{1}{3} x^{3})$

Step 1

Combine $x^{3}$ and $\frac{1}{3}$ .

$\frac{d}{d x} [x^{4} \ln (x - \frac{x^{3}}{3})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{4}$ and $g (x) = \ln (x - \frac{x^{3}}{3})$ .

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

Step 3

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 $g (x) = x - \frac{x^{3}}{3}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x - \frac{x^{3}}{3}$ .

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

Step 3.2

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

$x^{4} (\frac{1}{u} \frac{d}{d x} [x - \frac{x^{3}}{3}]) + \ln (x - \frac{x^{3}}{3}) \frac{d}{d x} [x^{4}]$

Step 3.3

Replace all occurrences of $u$ with $x - \frac{x^{3}}{3}$ .

$x^{4} (\frac{1}{x - \frac{x^{3}}{3}} \frac{d}{d x} [x - \frac{x^{3}}{3}]) + \ln (x - \frac{x^{3}}{3}) \frac{d}{d x} [x^{4}]$

Step 4

Differentiate.

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Step 4.1

Combine $\frac{1}{x - \frac{x^{3}}{3}}$ and $x^{4}$ .

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

Step 4.2

By the Sum Rule, the derivative of $x - \frac{x^{3}}{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{x^{3}}{3}]$ .

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

Step 4.3

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

$\frac{x^{4}}{x - \frac{x^{3}}{3}} (1 + \frac{d}{d x} [- \frac{x^{3}}{3}]) + \ln (x - \frac{x^{3}}{3}) \frac{d}{d x} [x^{4}]$

Step 4.4

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{x^{3}}{3}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{x^{4}}{x - \frac{x^{3}}{3}} (1 - \frac{1}{3} \frac{d}{d x} [x^{3}]) + \ln (x - \frac{x^{3}}{3}) \frac{d}{d x} [x^{4}]$

Step 4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 4.6

Simplify terms.

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Step 4.6.1

Multiply $3$ by $- 1$ .

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

Step 4.6.2

Combine $- 3$ and $\frac{1}{3}$ .

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

Step 4.6.3

Combine $\frac{- 3}{3}$ and $x^{2}$ .

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

Step 4.6.4

Cancel the common factor of $- 3$ and $3$ .

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Step 4.6.4.1

Factor $3$ out of $- 3 x^{2}$ .

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

Step 4.6.4.2

Cancel the common factors.

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Step 4.6.4.2.1

Factor $3$ out of $3$ .

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

Step 4.6.4.2.2

Cancel the common factor.

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

Step 4.6.4.2.3

Rewrite the expression.

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

Step 4.6.4.2.4

Divide $- x^{2}$ by $1$ .

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

Step 4.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{x^{4}}{x - \frac{x^{3}}{3}} (1 - x^{2}) + \ln (x - \frac{x^{3}}{3}) (4 x^{3})$

Step 4.8

Reorder terms.

$(1 - x^{2}) \frac{x^{4}}{x - \frac{x^{3}}{3}} + 4 x^{3} \ln (x - \frac{x^{3}}{3})$