Calculus Examples

$f (x) = \ln (x) + 2 e^{x} - 3 x^{2}$

Step 1

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

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

Step 2

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

$\frac{1}{x} + \frac{d}{d x} [2 e^{x}] + \frac{d}{d x} [- 3 x^{2}]$

Step 3

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

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Step 3.1

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

$\frac{1}{x} + 2 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 3 x^{2}]$

Step 3.2

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

$\frac{1}{x} + 2 e^{x} + \frac{d}{d x} [- 3 x^{2}]$

Step 4

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{x} + 2 e^{x} - 3 \frac{d}{d x} [x^{2}]$

Step 4.2

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

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

Step 4.3

Multiply $2$ by $- 3$ .

$\frac{1}{x} + 2 e^{x} - 6 x$

Step 5

Reorder terms.

$- 6 x + \frac{1}{x} + 2 e^{x}$