Calculus Examples

$f (x) = 1 + x + x^{2} + \ln (x)$

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $1 + x + x^{2} + \ln (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\ln (x)]$ .

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

Step 1.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\ln (x)]$

Step 1.1.3

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

$0 + 1 + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\ln (x)]$

Step 1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 + 1 + 2 x + \frac{d}{d x} [\ln (x)]$

Step 1.2

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

$0 + 1 + 2 x + \frac{1}{x}$

Step 1.3

Simplify.

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Step 1.3.1

Add $0$ and $1$ .

$1 + 2 x + \frac{1}{x}$

Step 1.3.2

Reorder terms.

$f' (x) = 2 x + \frac{1}{x} + 1$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $2 x + \frac{1}{x} + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [1]$

Step 2.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [1]$

Step 2.2.2

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

$2 \cdot 1 + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [1]$

Step 2.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [1]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 2.3.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$2 + \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [1]$

Step 2.3.2

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

$2 - x^{- 2} + \frac{d}{d x} [1]$

Step 2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 - x^{- 2} + 0$

Step 2.5

Simplify.

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Step 2.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 - \frac{1}{x^{2}} + 0$

Step 2.5.2

Add $2 - \frac{1}{x^{2}}$ and $0$ .

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

Step 2.5.3

Reorder terms.

$f'' (x) = - \frac{1}{x^{2}} + 2$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{1}{x^{2}} + 2$ .

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