Calculus Examples

$f (x) = 7 e^{- \frac{x}{3}}$

Step 1

Find the first derivative.

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Step 1.1

Since $7$ is constant with respect to $x$ , the derivative of $7 e^{- \frac{x}{3}}$ with respect to $x$ is $7 \frac{d}{d x} [e^{- \frac{x}{3}}]$ .

$7 \frac{d}{d x} [e^{- \frac{x}{3}}]$

Step 1.2

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 $g (x) = - \frac{x}{3}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $- \frac{x}{3}$ .

$7 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x}{3}])$

Step 1.2.2

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

$7 (e^{u} \frac{d}{d x} [- \frac{x}{3}])$

Step 1.2.3

Replace all occurrences of $u$ with $- \frac{x}{3}$ .

$7 (e^{- \frac{x}{3}} \frac{d}{d x} [- \frac{x}{3}])$

Step 1.3

Differentiate.

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Step 1.3.1

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{x}{3}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x]$ .

$7 e^{- \frac{x}{3}} (- \frac{1}{3} \frac{d}{d x} [x])$

Step 1.3.2

Combine fractions.

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Step 1.3.2.1

Multiply $- 1$ by $7$ .

$- 7 e^{- \frac{x}{3}} (\frac{1}{3} \frac{d}{d x} [x])$

Step 1.3.2.2

Combine $\frac{1}{3}$ and $- 7$ .

$\frac{- 7}{3} e^{- \frac{x}{3}} \frac{d}{d x} [x]$

Step 1.3.2.3

Combine $\frac{- 7}{3}$ and $e^{- \frac{x}{3}}$ .

$\frac{- 7 e^{- \frac{x}{3}}}{3} \frac{d}{d x} [x]$

Step 1.3.2.4

Move the negative in front of the fraction.

$- \frac{7 e^{- \frac{x}{3}}}{3} \frac{d}{d x} [x]$

Step 1.3.3

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

$- \frac{7 e^{- \frac{x}{3}}}{3} \cdot 1$

Step 1.3.4

Multiply $- 1$ by $1$ .

$f' (x) = - \frac{7 e^{- \frac{x}{3}}}{3}$

Step 2

Find the second derivative.

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Step 2.1

Since $- \frac{7}{3}$ is constant with respect to $x$ , the derivative of $- \frac{7 e^{- \frac{x}{3}}}{3}$ with respect to $x$ is $- \frac{7}{3} \frac{d}{d x} [e^{- \frac{x}{3}}]$ .

$- \frac{7}{3} \frac{d}{d x} [e^{- \frac{x}{3}}]$

Step 2.2

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 $g (x) = - \frac{x}{3}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $- \frac{x}{3}$ .

$- \frac{7}{3} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x}{3}])$

Step 2.2.2

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

$- \frac{7}{3} (e^{u} \frac{d}{d x} [- \frac{x}{3}])$

Step 2.2.3

Replace all occurrences of $u$ with $- \frac{x}{3}$ .

$- \frac{7}{3} (e^{- \frac{x}{3}} \frac{d}{d x} [- \frac{x}{3}])$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $e^{- \frac{x}{3}}$ and $\frac{7}{3}$ .

$- \frac{e^{- \frac{x}{3}} \cdot 7}{3} \frac{d}{d x} [- \frac{x}{3}]$

Step 2.3.2

Move $7$ to the left of $e^{- \frac{x}{3}}$ .

$- \frac{7 \cdot e^{- \frac{x}{3}}}{3} \frac{d}{d x} [- \frac{x}{3}]$

Step 2.3.3

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{x}{3}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x]$ .

$- \frac{7 e^{- \frac{x}{3}}}{3} (- \frac{1}{3} \frac{d}{d x} [x])$

Step 2.3.4

Combine fractions.

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{7 e^{- \frac{x}{3}}}{3} (\frac{1}{3} \frac{d}{d x} [x])$

Step 2.3.4.2

Multiply $\frac{7 e^{- \frac{x}{3}}}{3}$ by $1$ .

$\frac{7 e^{- \frac{x}{3}}}{3} (\frac{1}{3} \frac{d}{d x} [x])$

Step 2.3.4.3

Multiply $\frac{1}{3}$ by $\frac{7 e^{- \frac{x}{3}}}{3}$ .

$\frac{7 e^{- \frac{x}{3}}}{3 \cdot 3} \frac{d}{d x} [x]$

Step 2.3.4.4

Multiply $3$ by $3$ .

$\frac{7 e^{- \frac{x}{3}}}{9} \frac{d}{d x} [x]$

Step 2.3.5

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

$\frac{7 e^{- \frac{x}{3}}}{9} \cdot 1$

Step 2.3.6

Multiply $\frac{7 e^{- \frac{x}{3}}}{9}$ by $1$ .

$f'' (x) = \frac{7 e^{- \frac{x}{3}}}{9}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $\frac{7 e^{- \frac{x}{3}}}{9}$ .

$\frac{7 e^{- \frac{x}{3}}}{9}$