Calculus Examples

$f (x) = \ln (e^{3 x}) + x \ln (e^{2}) - 10 \ln (e)$

Step 1

Use logarithm rules to move $2$ out of the exponent.

$\frac{d}{d x} [\ln (e^{3 x}) + x (2 \ln (e)) - 10 \ln (e)]$

Step 2

The natural logarithm of $e$ is $1$ .

$\frac{d}{d x} [\ln (e^{3 x}) + x (2 \cdot 1) - 10 \ln (e)]$

Step 3

Multiply $2$ by $1$ .

$\frac{d}{d x} [\ln (e^{3 x}) + x \cdot 2 - 10 \ln (e)]$

Step 4

The natural logarithm of $e$ is $1$ .

$\frac{d}{d x} [\ln (e^{3 x}) + x \cdot 2 - 10 \cdot 1]$

Step 5

By the Sum Rule, the derivative of $\ln (e^{3 x}) + x \cdot 2 - 10 \cdot 1$ with respect to $x$ is $\frac{d}{d x} [\ln (e^{3 x})] + \frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- 10 \cdot 1]$ .

$\frac{d}{d x} [\ln (e^{3 x})] + \frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- 10 \cdot 1]$

Step 6

Evaluate $\frac{d}{d x} [\ln (e^{3 x})]$ .

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Step 6.1

Use logarithm rules to move $3 x$ out of the exponent.

$\frac{d}{d x} [3 x \ln (e)] + \frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- 10 \cdot 1]$

Step 6.2

The natural logarithm of $e$ is $1$ .

$\frac{d}{d x} [3 x \cdot 1] + \frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- 10 \cdot 1]$

Step 6.3

Multiply $3$ by $1$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- 10 \cdot 1]$

Step 6.4

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- 10 \cdot 1]$

Step 6.5

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

$3 \cdot 1 + \frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- 10 \cdot 1]$

Step 6.6

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- 10 \cdot 1]$

Step 7

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

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Step 7.1

Move $2$ to the left of $x$ .

$3 + \frac{d}{d x} [2 \cdot x] + \frac{d}{d x} [- 10 \cdot 1]$

Step 7.2

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

$3 + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 10 \cdot 1]$

Step 7.3

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

$3 + 2 \cdot 1 + \frac{d}{d x} [- 10 \cdot 1]$

Step 7.4

Multiply $2$ by $1$ .

$3 + 2 + \frac{d}{d x} [- 10 \cdot 1]$

Step 8

Evaluate $\frac{d}{d x} [- 10 \cdot 1]$ .

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Step 8.1

Multiply $- 10$ by $1$ .

$3 + 2 + \frac{d}{d x} [- 10]$

Step 8.2

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$3 + 2 + 0$

Step 9

Combine terms.

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Step 9.1

Add $3$ and $2$ .

$5 + 0$

Step 9.2

Add $5$ and $0$ .

$5$