Calculus Examples

$y = \ln (e^{7 x} + 2)$

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 $g (x) = e^{7 x} + 2$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $e^{7 x} + 2$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [e^{7 x} + 2]$

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [e^{7 x} + 2]$

Step 1.3

Replace all occurrences of $u_{1}$ with $e^{7 x} + 2$ .

$\frac{1}{e^{7 x} + 2} \frac{d}{d x} [e^{7 x} + 2]$

Step 2

By the Sum Rule, the derivative of $e^{7 x} + 2$ with respect to $x$ is $\frac{d}{d x} [e^{7 x}] + \frac{d}{d x} [2]$ .

$\frac{1}{e^{7 x} + 2} (\frac{d}{d x} [e^{7 x}] + \frac{d}{d x} [2])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 7 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

$\frac{1}{e^{7 x} + 2} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [7 x] + \frac{d}{d x} [2])$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$\frac{1}{e^{7 x} + 2} (e^{u_{2}} \frac{d}{d x} [7 x] + \frac{d}{d x} [2])$

Step 3.3

Replace all occurrences of $u_{2}$ with $7 x$ .

$\frac{1}{e^{7 x} + 2} (e^{7 x} \frac{d}{d x} [7 x] + \frac{d}{d x} [2])$

Step 4

Differentiate.

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Step 4.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{1}{e^{7 x} + 2} (e^{7 x} (7 \frac{d}{d x} [x]) + \frac{d}{d x} [2])$

Step 4.2

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

$\frac{1}{e^{7 x} + 2} (e^{7 x} (7 \cdot 1) + \frac{d}{d x} [2])$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $7$ by $1$ .

$\frac{1}{e^{7 x} + 2} (e^{7 x} \cdot 7 + \frac{d}{d x} [2])$

Step 4.3.2

Move $7$ to the left of $e^{7 x}$ .

$\frac{1}{e^{7 x} + 2} (7 \cdot e^{7 x} + \frac{d}{d x} [2])$

Step 4.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1}{e^{7 x} + 2} (7 e^{7 x} + 0)$

Step 4.5

Combine fractions.

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Step 4.5.1

Add $7 e^{7 x}$ and $0$ .

$\frac{1}{e^{7 x} + 2} (7 e^{7 x})$

Step 4.5.2

Combine $7$ and $\frac{1}{e^{7 x} + 2}$ .

$\frac{7}{e^{7 x} + 2} e^{7 x}$

Step 4.5.3

Combine $\frac{7}{e^{7 x} + 2}$ and $e^{7 x}$ .

$\frac{7 e^{7 x}}{e^{7 x} + 2}$