Calculus Examples

$\ln (3 + x - x^{3})$

Step 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 $g (x) = 3 + x - x^{3}$ .

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Step 1.1

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

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

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [3 + x - x^{3}]$

Step 1.3

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

$\frac{1}{3 + x - x^{3}} \frac{d}{d x} [3 + x - x^{3}]$

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{1}{3 + x - x^{3}} (0 + \frac{d}{d x} [x] + \frac{d}{d x} [- x^{3}])$

Step 2.3

Add $0$ and $\frac{d}{d x} [x]$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $3$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

Reorder the factors of $\frac{1}{3 + x - x^{3}} (1 - 3 x^{2})$ .

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

Step 3.2

Multiply $1 - 3 x^{2}$ by $\frac{1}{3 + x - x^{3}}$ .

$\frac{1 - 3 x^{2}}{3 + x - x^{3}}$