Calculus Examples

$f (x) = 2 e^{x} + 5 x - \ln (x)$

Step 1

By the Sum Rule, the derivative of $2 e^{x} + 5 x - \ln (x)$ with respect to $x$ is $\frac{d}{d x} [2 e^{x}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- \ln (x)]$ .

$\frac{d}{d x} [2 e^{x}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- \ln (x)]$

Step 2

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

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Step 2.1

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

$2 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- \ln (x)]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$2 e^{x} + \frac{d}{d x} [5 x] + \frac{d}{d x} [- \ln (x)]$

Step 3

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$2 e^{x} + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- \ln (x)]$

Step 3.2

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

$2 e^{x} + 5 \cdot 1 + \frac{d}{d x} [- \ln (x)]$

Step 3.3

Multiply $5$ by $1$ .

$2 e^{x} + 5 + \frac{d}{d x} [- \ln (x)]$

Step 4

Evaluate $\frac{d}{d x} [- \ln (x)]$ .

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Step 4.1

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

$2 e^{x} + 5 - \frac{d}{d x} [\ln (x)]$

Step 4.2

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

$2 e^{x} + 5 - \frac{1}{x}$

Step 5

Reorder terms.

$- \frac{1}{x} + 2 e^{x} + 5$